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r
KPl-e^O^OCi)
HARVARD COLLEGE LIBRARY
k
THE THIRTEEN BOOK3
OF
EUCLID'S ELEMENTS
CAMBRIDGE UNIVERSITY PRESS WAREHOUSE,
C. F. CLAY, Manager.
aottlroii: FETTER LANE, E.G.
fftlfotorfft: >«>> PRINCES STREET.
^m
Serlfo: A. ASHER AND CO.
Inpjifl: F. A. BROCKHAUS.
|Uf» Sorfc: G. P. PUTNAM'S SONS.
Vombig mH ffalrutta: MACMILLAN AND CO., Ltd.
[AU Rights reserved.'^
THE THIRTEEN BOOKS
OF
EUCLID'S ELEMENTS
TRANSLATED FROM THE TEXT OF HEIBERG
WITH INTRODyCTION AND COMMENTARY
BY T. L. HEATH, C.B., Sc.D.,
SOMETIME FELLOW OF TRINITY COLLEGE, CAMBRIDGE
VOLUME IN BOOKS X— XIII AND APPENDIX
Cambridge : at the University Press
1908
/
Mt?.x/.- rf?H:-^X
-er-f—^—^
. y fiarrord UnJwrrthr ^ Math. Depl. Ubraqi
Cambittgr:
PRINTED BY JOHN CLAY, M.A. AT TKB UNIVEKStTY PRESS.
CONTENTS OF VOLUME III
PAGB
Book X. Introductory note i
Definitions lo
Propositions i — 47 . . . . . . 14-101
Definitions ii. . 101
Propositions 48 — 84 102-177
Definitions hi 177
Propositions 85 — 115 178-254
Ancient extensions of theory of Book X . 255
Book XI. Definitions 260
Propositions 272
Book XIL Historical note 365
Propositions 369
Book XIII. Historical note 438
Propositions 440
Appendix. I. The so-called "Book XIV." (by Hypsicles) 512
II. Note on the so-called "Book XV." . . 5x9
Addenda et Corrigenda 521
General Index: Greek 529
„ „ English 535
r
BOOK X.
INTRODUCTORV NOTE.
The discovery of the doctrine of incommensurables is attributed to Pythagoras. Thus Proclus says (Comtn, on EucL i. p. 65, 19) that Pythagoras "discovered the theory of irrationals*"; and, again, the scholium on the begin- ning of Book X., also attributed to Proclus, states that the Pythagoreans were the first to address themselves to the investigation of commensurability, having discovered it by means of their observation of numbers. They discovered, the scholium continues, that not all magnitudes have a common measure. " They called all magnitudes measurable by the same measure commensurable, but those which are not subject to the same measure incommensurable, and again such of these as are measured by some other common measure commensurable with one another, and such as are not, incommensurable with the others. And thus by assuming their measures they referred everything to different commensurabilities, but, though they were different, even so (they proved that) not all magnitudes are commensurable with any. (They showed that) all magnitudes can be rational (pi^ra) and all irrational (aAoya) in a relative sense (ok vpos n); hence the commensurable and the incommensurable would be for them naturcU (kinds) (i^vaci), while the rational and irrational would rest on assumption or convention (Oia-ei)" The scholium quotes further the l^end according to which " the first of the Pythagoreans who made public the investigation of these matters perished in a shipwreck," conjecturing that the authors of this story " perhaps spoke allegorically, hinting that everything irrational and formless is properly concealed, and, if any soul should rashly invade this region of life and lay it open, it would be carried away into the sea of becoming and be overwhelmed by its unresting currents." There would be a reason also for keeping the discovery of irrationals secret for the time in the fact that it rendered unstable so much of the groundwork of geometry as the Pythagoreans had based upon the imperfect theory of proportions which applied only to numbers. We have already, after Tannery, referred to the probability that the discovery of incommensurability must have necessitated a great recasting of the whole fabric of elementary geometry, pending the discovery of the general theory of proportion applicable to incommensurable as well as to commensurable magnitudes.
It seems certain that it was with reference to the length of the diagonal of a square or the hypotenuse of an isosceles right-angled triangle that Pythagoras made his discovery. Plato (TheaetetuSy 147 d) tells us that Theodorus of Cyrene wrote about square roots (SuFa/xcis), proving that the square roots of
* I have already noted (Vol. i. p. 351) that G. Junge {Wann hahen die Griechen das IrratumaU entdecktf) disputes this, maintaining that it was the Pythagoreans, but not Pythagoras, who made the discovery. Junge is obliged to alter the reading of the passage of Proclus, on what seems to be quite insufficient evidence ; and in any case I doubt whether the point is worth so much labouring.
H. E. III. . I
vi-^
2 BOOK X
three square feet and five square feet are not commensurable with that of one square foot, and so on, selecting each such square root up to that of 1 7 square feet, at which for some reason he stopped. No mention is here made of J2y doubtless for the reason that its incommensurability had been proved before^ i.e. by Pythagoras. We know that Pythagoras invented a formula for finding right-angled triangles in rational numbers, and in connexion with this it was inevitable that he should investigate the relations between sides and hypotenuse in other right-angled triangles. He would naturally give special attention to the isosceles right-angled triangle ; he would try to measure the diagonal, he would arrive at successive approximations, in rational fractions, to the value of J2 ; he would find that successive efforts to obtain an exact expression for it failed. It was however an enormous step to conclude that such exact expression was impossible^ and it was this step which Pythagoras (or the Pythagoreans) made. We now know that the formation of the side- and ^//Vi^;/a/-numbers explained by Theon of Smyrna and others was Pythagorean, and also that the theorems of Eucl. 11. 9, 10 were used by the Pythagoreans in direct connexion with this method of approximating to the value of ^2. The very method by which Euclid proves these propositions is itself an indica- tion of their connexion with the investigation of ^2, since he uses a figure made up of two isosceles right-angled triangles.
The actual method by which the Pythagoreans proved the incommensura- bility of »J2 with unity was no doubt that referred to by Aristotle {Anal, prior. 1. 23, 4 1 a 26 — 7 ), a reductio adabsurdum by which it is proved that, if the diagonal is commensurable with the side, it will follow that the same number is both odd and even. The proof formerly appeared in the texts of Euclid as x. 117, but it is undoubtedly an interpolation, and August and Heiberg accordingly relegate it to an Appendix. It is in substance as follows.
Suppose AC^ the diagonal of a square, to be commen- surable with AB^ its side. Let a : j8 be their ratio expressed in the smallest numbers.
Then a > j8 and therefore necessarily > i.
Now AC^.AB^^a^xP",
and, since AC^=2AB*, [Eucl. i. 47]
a» = 2)8*.
Therefore a' is even, and therefore a is even.
Since a : j8 is in its lowest terms, it follows that P must be odd
Put a=2y;
therefore 4y* = 2j8*,
or p»= 2/,
so that j8*, and therefore /8, must be ei^en.
But P was also odd : which is impossible.
This proof only enables us to prove the incommensurability of the diagonal of a square with its side, or of J2 with unity. In order to prove I he incommensurability of the sides of squares, one of which has lAree times the area of another, an entirely different procedure is necessary ; and we find in fact that, even a century after Pythagoras' time, it was still necessary to use separate proofs (as the passage of the Theaeteius shows that Theodorus did) to establish the incommensurability with unity of ^3, ^5, ... up to ^17.
INTRODUCTORY NOTE 3
This fact indicates clearly that the general theorem in Eucl. x. 9 that squares which have not to one another the ratio of a square number to a square number have thdr sides ificommensurable in length was not arrived at all at once, but was, in the manner of the time, developed out of the separate consideration of special cases (Hankel, p. 103).
The proposition x. 9 of Euclid is definitely ascribed by the scholiast to Theaetetus. Theaetetus was a pupil of Theodorus, and it would seem clear that the theorem was not known to Theodorus. Moreover the Platonic passage itself {Theaet 147 d sqq.) represents the young Theaetetus as striving after a general conception of what we call a surd, " The idea occurred to me, seeing that square roots (Woftcts) appeared to be unlimited in multitude, to try to arrive at one collective term by which we could designate all these square roots — I divided number in general into two classes. The number which can be expressed as equal multiplied by equal (mtov Wkis) I likened to a square in form, and I called it square and equilateral.... The intermediate number, such as three, five, and any number which cannot be expressed as equal multiplied by equal, but is either less times more or more times less, so that it is always contained by a greater and less side, I likened to an oblong figure and called an oblong number. ...Such straight lines then as square the equilateral and plane number I defined as length (fi^icos), and such as square the oblong square roots (^ii^a/icis), as not being commensurable with the others in length but only in the plane areas to which their squares are equal. "
There is further evidence of the contributions of Theaetetus to the theory of incommensurables in a commentary on Eucl. x. discovered, in an Arabic translation, by Woepcke {Mimoires prisenths d tAcadkmie des Sciences^ xiv., 1^56, pp. 658 — 720). It is certain that this commentary is of Greek origin. Woepcke conjectures that it was by Vettius Valens, an astronomer, apparently of Antioch; and a contemporary of Claudius Ptolemy (2nd cent. A.D.). Heiberg, with greater probability, thinks that we have here a fragment of the commentary of Pappus (Euklid-studien^ pp. 169 — 71), and this is rendered practically certain by Suter (Z>/> Mathematiker und Astronomen der Araber und ihre Werke^ pp. 49 and 211). This commentary states that the theory of irrational magnitudes *' had its origin in the school of Pythagoras. It was considerably developed by Theaetetus the Athenian, who gave proof, in this part of mathematics, as in others, of ability which has been justly admired. He was one of the most happily endowed of men, and gave himself up, with a fine enthusiasm, to the investigation of the truths contained in these sciences, as Plato bears witness for him in the work which he called after his name. As for the exact distinctions of the above-named magnitudes and the rigorous demonstrations of the propositions to which this theory gives rise, I believe that they were chiefly established by this mathematician; and, later, the great ApoUonius, whose genius touched the highest point of excellence in mathematics, added to these discoveries a number of remarkable theories after many efforts and much labour.
"For Theaetetus had distinguished square roots [puissances must be the Swafici« of the Platonic passage] commensurable in length from those which are incommensurable, and had divided the well-known species of irrational lines after the different means, assigning the medial to geometry, the binomial to arithmetic, and the apotome to harmony, as is stated by Eudemus the Peripatetic.
" As for Euclid, he set himself to give rigorous rules, which he established.
4 BOOK X
relative to commensurability and incommensurability in general; he made precise the definitions and the distinctions between rational and irrational magnitudes, he set out a great number of orders of irrational magnitudes, and finally he clearly showed their whole extent"
The allusion in the last words must be apparently to x. 115, where it is proved that from the medial straight line an unlimited number of other irrationals can be derived all different from it and from one another.
The connexion between the medial straight line and the geometric mean is obvious, because it is in fact the mean proportional between two rational straight lines ''commensurable in square only." Since ^(X'^y) is the arithmetic mean between x^ y^ the reference to it of the binomial can be understood. The connexion between the apotome and the harmonic mean is explained by some propositions in the second book of the Arabic commentary. The
harmonic mean between x, y is — - , and propositions of which Woepcke
quotes the enunciations prove that, if a rational or a medial area has for one of its sides a binomial straight line, the other side will be an apotome of corre- sponding order (these propositions are generalised from Eucl. x. in — 4); the
fact is that ^^ = -^^ . ix -y\ x+y x^-y ^ -^^
One other predecessor of Euclid appears to have written on irrationals, though we know no more of the work than its title as handed down by Diogenes Laertius^ According to this tradition, Democritus wrote ircpi oAoyoiv ypafifxoiv Koi vaariuv jS', two Books on irrational straight lines and solids (apparently). Hultsch {Neue Jahrbiicher fiir Philologie und Padagogik^ 1881, pp. 578 — 9) conjectures that the true reading may be ircpl aXoyioy ypafifji^v KkaurrSv, "on irrational broken lines." Hultsch seems to have m mind straight lines divided into two parts one of which is rational and the other irrational (''Aus einer Art von Umkehr des Pythagoreischen Lehrsatzes iiber das rechtwinklige Dreieck gieng zunachst mit Leichtigkeit hervor, dass man eine Linie construiren konne, welche als irrational zu bezeichnen ist, aber durch Brechung sich darstellen lasst als die Summe einer rationalen und einer irrationalen Linie"). But I doubt the use of #cXacrro« in the sense of breaking one straight line into parts ; it should properly mean a bent line, Le. two straight lines forming an angle or broken short off dX their point of meeting. It is also to be observed that vaarriv is quoted as a Democritean word (opposite to kcvw) in a fragment of Aristotle (202). I see therefore no reason for questioning the correctness of the title of Democritus' book as above quoted.
I will here quote a valuable remark of Zeuthen's relating to the classifi- cation of irrationals. He says (Geschichte der Mathematik im Altertum und Mittelaltery p. 56) "Since such roots of equations of the second degree as are incommensurable with the given magnitudes cannot be expressed by means of the latter and of numbers, it is conceivable that the Greeks, in exact investigations, introduced no approximate values but worked on with the magnitudes they had found, which were represented by straight lines obtained by the construction corresponding to the solution of the equation. That is exactly the same thing which happens when we do not evaluate roots but content ourselves with expressing them by radical signs and other algebraical symbols. But, inasmuch as one straight line looks like another, the Greeks did not get
^ Diog. Laert. ix. 47, p. 339 (ed. Cobet).
INTRODUCTORY NOTE 5
the same clear view of what they denoted (i.e. by simple inspection) as our system of symbols assures to us. For this reason it was necessary to under- take a classification of the irrational magnitudes which had been arrived at by successive solution of equations of the second degree." To much the same effect Tannery wrote in 1882 {De la solution ghmetrique des problemes du second degre avani Euclide in Mhnoires de la Sociitk des sciences physiques et naturelles de Bordeaux^ 2* Serie, iv. pp. 395—416). Accordingly Book x. formed a repository of results to which could be referred problems which depended on the solution of certain types of equations, quadratic and biquad- ratic but reducible to quadratics. Consider the quadratic equations
Jc'±2flUitr.p + ^.p» = o,
where p is a rational straight line, and a, fi are coefficients. Our quadratic equations in algebra leave out the p ; but I put it in, because it has always to be remembered that Euclid's ^ is a straight line, not an algebraical quantity, and is therefore to be found in terms of, or in relation to, a certain assumed rational straight line^ and also because with Euclid p may be not only of the
form a, where a represents a units of length, but also of the form fj — . a,
which represents a length "commensurable in square only" with the unit of length, or ^A where A represents a number (not square) of units of area. The use therefore of p in our equations makes it unnecessary to multiply different cases according to the relation of p to the unit of length, and has the further advantage that, e.g., the expression p±Jh,pis just as general as the expression s/k,p±J\.py since p covers the form Jk . p, both expressions covering a length either commensurable in length, or "commensurable in square only," with the unit of length.
Now the positive roots of the quadratic equations
jc" + 2ar . p ± /5 . p' = o
can only have the following forms
^, = p(a + Va^),< = p(a-\/c?^) |
.r, = p ( Va* + fi + a), Xi = p (Va* + )8 - o) / *
The negative roots do not come in, since x must be a straight line. The omission however to bring in negative roots constitutes no loss of generality, since the Greeks would write the equation leading to negative roots in another form so as to make them positive, i.e. they would change the sign of x in the equation.
Now the positive roots or,, .r,', x^, x^ may be classified according to the character of the coefficents a, fi and their relation to one another.
I. Suppose that a, p do not contain any surds, i.e. are cither integers or of the form mjn^ where /«, n are integers.
Now in the expressions for jrj, jc/ it may be that
(1) /9 is of the form -^ o'.
Euclid expresses this by saying that the square on op exceeds the square on pJor — p by the square on a straight line commensurable in length with op. In this case x^ is, in Euclid's terminology, k first dinomial straight line, and Xi a first apotome.
BOOK X
(2) In general, fi not being of the form -5^ a',
jTi is 9^ fourth binomicU^ Xi di fourth apotome.
Next, in the expressions for jc^, jr,' it may be that
(i) P is equal to — ^ (o' + /8), where m, n are integers, i.e. P is of the form
Euclid expresses this by saying that the square on ps/a' + jS exceeds the square on op by the square on a straight line commensurable in length with
In this case x^ is, in Euclid's terminology, a second binomial^ x^ a second apotome,
(2) In general, ^ not being of the form -j-^^ ,a*,
x<^ is 9i fifth binomial^ xi di fifth apotome.
y^.wh
II. Now suppose that a is of the form ^ — , where »i, n are integers, and
let us denote it by ^X, Then in this case
Thus jfi, xj are of the same form as ^, x^.
If n/X - ^ in x^ , xi is not surd but of the form m/n, and if J\ + fi in Xj, xi is not surd but of the form m/n, the roots are comprised among the forms already shown, the first, second, fourth and fifth binomials and apotomes. If JX-p in Xj, xi is surd, then
(i) we may have fi of the form -^ A, and in this case
H
Xi is a third binomai straight line, xi a third apotome\
(2) in general, p not being of the form ^ A,
Xx is a sixth binomial straight line, xi a sixth apotome. With the expressions for x^, xi the distinction between the third and sixth bmomials and apotomes is of course the distinction between the cases
(i) in which ^8= ^ (X + ^), or /8 is of the form ~^^K « n*- m*
and (2) in which fi is not of this form.
If we take the square root of the product of p and each of the six
binomials and six apotomes just classified, i.e.
INTRODUCTORY NOTE 7
in the six different forms that each may take, we find six new irrationals with a positive sign separating the two terms, and six corresponding irrationals with a negative sign. These are of course roots of the equations
x*±2ax*,p*±p.p*=o. These irrationals really come before the others in Euclid's order (x. 36 — 41 for the positive sign and x. 73 — 78 for the negative sign). As we shall see in due course, the straight lines actually found by Euclid are
1. P± J^'Pi the binomial {yj ix Svo dvo/iaroiv)
and the apotome (airoro/AiJ), which are the positive roots of the biquadratic (reducible to a quadratic) :c*-2(i+>t)^.:i:»+(i->t)V* = o.
2. )^p ± k^p, iht first bitnedial (Ik Svo /acVoiv vpwnf) and the first apotome of a medial (jji€inf9 airoro/AiJ irpurn/),
which are the positive roots of
x'-2^k(i-^k)p*,x'-^k{i''kyp^ = o.
3. ^^p± ^ Py the second bimedial {U hvo fiiawv SevTcpo)
and the second apotome of a medial {lUxrq^ dworofii^ ^cvr^), which are the positive roots of the equation
pi ^ ^ P I ^ _
the major (irrational straight line) (fJAHtav) and the minor (irrational straight line) (cXoiro-iui'), which are the positive roots of jthe equation
a^-2p^.x^+ 7i p^ = o.
V2(l+^) V2(l+^) '
the *^side "ofa rational plus a medial (area) {jnjfTov #cat /ucVoi^ Svmficn;) and the ^^side^^ of a medial minus a rational area (in the Greek 17 ficra ^ov pAirov TO oAov irotovaa),
which are the positive roots of the equation
the **side" of the sum of two medial areas {rj 8vo ^e<ra hvvapivri) and the " side ^^ of a medial minus a medial area (in the Greek 17 /icra pAirov piaov TO oXov iroiovaa), which are the positive roots of the equation
X*-2j\. ^p'+ X jzp^ = o.
8 BOOK X
The above facts and formulae admit of being stated in a great variety of ways according to the notation and the particular letters used. Consequently the summaries which have been given of Eucl. x. by various writers differ much in appearance while expressing the same thing in substance. The first summary in algebraical form (and a very elaborate one) seems to have been that of Cossali (Ortginc, trasporto in Italia^ primi progressi in essa dfiP Algebra^ Vol. ii. pp. 242 — 65) who takes credit accordingly (p. 265). In 1794 Meier Hirsch published at Beriin an Algebraischer Commentar iiber das %ehente Buck der Elemente des Euklides which gives the contents in algebraical form but fails to give any indication of Euclid^s methods, using modem forms of proof only. In 1834 Poselger wrote a paper, Ueber das zehnte Buck dtr Elemente des Euklides, in which he pointed out the defects of Hirsch's repro- duction and gave a summary of his own, which however, though nearer to Euclid's form, is difficult to follow in consequence of an elaborate system of abbreviations, and is open to the objection that it is not algebraical enough to enable the character of Euclid's irrationals to be seen at a glance. Other summaries will be found (i) in Nesselmann, Die Algebra der Griecfun^ pp. 165 — 84; (2) in Loria, II periodo aureo delta geometria greca, Modena, 1895, pp. 40 — 9; (3) in Christensen's article "Ueber Gleichungen vierten Grades im zehnten Buch der Elemente Euklids" in the Zeitschrift fiir Afatk. u. Physik (Jfistorisch-literarische Abtheilung), xxxiv. (1889), pp. 201 — 17. The only summary in English that I know is that in the Penny Cyclopaedia^ under "Irrational quantity," by De Morgan, who yielded to none in his admiration of Book X. "Euclid investigates," says De Morgan, "every possible variety of lines which can be represented by »J{[ja + ^b\ a and b representing two commen- surable lines.... This book has a completeness which none of the others (not even the fifth) can boast of ; and we could almost suspect that Euclid, having arranged his materials in his own mind, and having completely elaborated the 10th Book, wrote the preceding books after it and did not live to revise them thoroughly."
Much attention was given to Book x. by the early algebraists. Thus Leonardo of Pisa (fi. about 1200 a.d.) wrote in the 14th section of his Liber Abaci on the theory of irrationalities {de tractatu binomiorum et recisorum\ without however (except in treating of irrational trinomials and cubic irra- tionalities) adding much to the substance of Book x.; and, in investigating the equation
propounded by Johannes of Palermo, he proved that none of the irrationals in Eucl. X. would satisfy it (Hankel, pp. 344—6, Cantor, iij, p. 43). Luca Paciuolo (about 1445 — 15 14 a.d.) in his algebra based himself largely, as he himself expressly says, on Euclid x. (Cantor, 11,, p. 293). Michael Stifel (1486 or 1487 to 1567) wrote on irrational numbers in the second Book of his Arithmetica Integra, which Book may be regarded, says Cantor (iij, p. 402), as an elucidation of Eucl. x. The works of Cardano (1501—76) abound in speculations regarding the irrationals of Euclid, as may be seen by reference to Cossali (Vol. II., especially pp. 268—78 and 382—99); the character of the various odd and even powers of the binomials and apotomes is therein investigated, and Cardano considers in detail of what particular forms of equations, quadratic, cubic, and biquadratic, each class of Euclidean irrationals can be roots. Simon Stevin ( 1 548— 1620) wrote a TraitS des incommensuradles grandeurs en laquelle est sommairement diclarh le contenu du Dixiesme Livre d!Euclide(Peuvres mathimatiques, Leyde, 1634, pp. 219 sqq.); he speaks thus
INTRODUCTORY NOTE 9
of the book : *' La difficulte du dixiesme Livre d'Euclide est k plusieurs devenue en horreur, voire j usque k Tappeler la croix des mathematiciens, mati^re trop dure k digerer, et en la quelle n'aper9oivent aucune utilite," a passage quoted by I^ria i^Jl periodo aureo delta geomeiria greca^ p. 41).
It will naturally be asked, what use did the Greek geometers actually make of the theory of irrationals developed at such length in Book x. ? The answer is that Euclid himself, in Book xiii., makes considerable use of the second {portion of Book x. dealing with the irrationals affected with a negative sign, the apotomes etc. One object of Book xiii. is to investigate the relation of the sides of a pentagon inscribed in a circle and of an icosahedron and dodecahedron inscribed in a sphere to the diameter of the circle or sphere respectively, supposed rational. The connexion with the regular pentagon of a straight line cut in extreme and mean ratio is well known, and Euclid first proves (xiii. 6) that, if a rational straight line is so divided, the parts are the irrationals called apotomes, the lesser part being a first apotome. Then, on the assumption that the diameters of a circle and sphere respectively are rational, he proves (xui. 11) that the side of the inscribed regular pentagon is the irrational straight line called minora as is also the side of the inscribed icosahedron (xiii. 16), while the side of the inscribed dodecahedron is the irrational called an apotome (xiii. 17).
Of course the investigation in Book x. would not have been complete if it had dealt only with the irrationals affected with a negative sign. Those affected with the positive sign, the binomials etc, had also to be discussed, and we find both portions of Book x., with its nomenclature, made use of by Pappus in two propositions, of which it may be of interest to give the enun- ciations here.
If, says Pappus (iv. p. 178), AB be the rational diameter of a semicircle, and if A B he produced to C so that BC is equal to the radius, if C£> be a tangent,
if ^ be the middle point of the arc B£>y and if C£ be joined, then C£ is the irrational straight line called minor. As a matter of fact, if p is the radius,
r^ = p'(5-«V3) and C£=Js+^ _ Vsji^y .
V 2 ▼ 2
If, again (p. 182), CD be equal to the radius of a semicircle supposed
rational, and if the tangent £>B be drawn and the angle AUB be bisected by DF meeting the circumference in F, then DF is the excess by which the binomial exceeds the straight line which produces with a rational area a medial
lO BOOK X [X. DEFF. 1—4
whole (see EucL x. 77). (In the figure DK\& the dinomia/ and KFthe other irrational straight line.) As a matter of fact, if p be the radius,
Proclus tells us that Euclid left out, as alien to a selection of elenunts, the discussion of the more complicated irrationals, ''the unordered irrationals which Apollonius worked out more fully" (Proclus, p. 74, 23), while the scholiast to Book X. remarks that Euclid does not deal with all rationals and irrationals but only the simplest kinds by the combination of which an infinite number of irrationals are obtained, of which Apollonius also gave some. The author of the commentary on Book x. found by Woepcke m an Arabic translation, and above alluded to, also says that "it was Apollonius who, beside the ordered irrational magnitudes, showed the existence of the unordered and by accurate methods set forth a great number of them." It can only be vaguely gathered, from such hints as the commentator proceeds to give, what the character of the extension of the subject given by Apollonius may have been. See note at end of Book.
DEFINITIONS.
1. Those magnitudes are said to be commensurable which are measured by the same measure, and those incom- mensurable which cannot have any common measure.
2. Straight lines are commensurable in square when the squares on them are measured by the same area, and incommensurable in square when the squares on them cannot possibly have any area as a common measure.
3. With these hypotheses, it is proved that there exist straight lines infinite in multitude which are commensurable and incommensurable respectively, some in length only, and others in square also, with an assigned straight line. Let then the assigned straight line be called rational, and those straight lines which are commensurable with it, whether in length and in square or in square only, rational, but those which are incommensurable with it irrational.
4. And let the square on the assigned straight line be called rational and those areas which are commensurable with it rational, but those which are incommensurable with it irrational, and the straight lines which produce them irrational, that is, in case the areas are squares, the sides themselves, but in case they are any other rectilineal figures, the straight lines on which are described squares equal to them.
X. DEFF. 1—3] DEFINITIONS AND NOTES
Definition i.
^vfJLfierpa fjLiytOrf Acycrai ra r^ avrw /xcrpii) fierpovfJLeyii, d4rv/JLfi€Tpa 3c, cuf
Definition 2.
Ev^ciai SvFafici avfifierpot curiv, orai' ra dw' avraiy Ttrpdytaya t«J avr<p X^'V^ ft€TfnJTatj aav/ifMTpoi 3^ orai^ rocs dir' avr«Si' rrrpaycoi^ocs firfStv cvSc^vyTai \(u>piov Koivw fi€Tpov y€V€a'$ai.
Commensurabie in square is in the Greek Bwafi€t av/ifjivrpo^. In earlier translations (e.g. Williamson's) 3vi a/Aci has been translated " in power," but, as the particular power represented by 3uvafics in Greek geometry is square^ I have thought it best to use the latter word throughout. It will be observed that Euclid's expression commensurable in square only (used in Def. 3 and constantly) corresponds to what Plato makes Theaetetus call a square root (Svwi/us) in the sense of a surd. If a is any straight line, a and ajm, or a^m and a^n (where m^ n are integers or arithmetical fractions in their lowest terms, proper or improper, but not square) are commensurable in square only. Of course (as explained in the Porism to x. 10) all straight lines commensurable in length {/iiiKti), in Euclid's phrase, are commensurable in square also ; but not all straight lines which are commensurable in square are commensurable in length as well. On the other hand, straight lines incom- mensurable in square are necessarily incommensurable in length also \ but not all straight lines which are incommensurable in length are incommensurable in square. In fact, straight lines which are commensurable in square only are incommensurable in lengthy but obviously not incommensurable in square.
Definition 3.
TovTttiv viTOKccficVfitfi^ SctKiomii, 5ri tjJ irpoTc^can/ tx^w^. xnrdpxovanv €v$€uu vK^u dircipot avfifA€rpoi re #cai aa-v/ifierpoi al fAtv fii/fcci fwyov^ al 8c koi Svvdf/i€L JcoA.ciO'^io oivv 17 ficv ir/oorc^cc(ra cv^ctia pr/rq, ical at ravTy <rvfifJL€TpOL ctrc /xi/kci koi dvKa/iCi ctrc SwdfJL€i yuovov pntfrax^ aX Sc ravry davfifitTpoi dXoyoi #caAcicr^<iKrai'.
The first sentence of the definition is decidedly elliptical. It should, strictly speaking, assert that '' with a given straight line there are an infinite number of straight lines which are (i) commensurable either (a) in square only or {b) in square and in length also, and (2) incommensurable, either {a) in length only or {b) in length and in square also."
The relativity of the terms rational and irrational is well brought out in this definition. We may set out any straight line and call it rational, and it is then with reference to this assumed rational straight line that others are called rational or irrational.
We should carefully note that the signification of rational in Euclid is wider than in our terminology. With him, not only is a straight line commensurable in length with a rational straight line rational, but a straight line is rational which is commensurable with a rational straight line in square only. That is, if p is a
rational straight line, not only is — p rational, where m^ n are integers and
n
12 BOOK X [X. DEFF. 3, 4
mjn in its lowest terms is not square, but a/ — . p is rational also. We should
in this case call f^j — . p irrational. It would appear that Euclid's termino- logy here differed as much from that of his predecessors as it does from ours. We are familiar with the phrase appvfToq Sta/xcrpo^ t^? vtfjLirdSfK by which Plato (evidently after the Pythagoreans) describes the diagonal of a square on a straight line containing 5 units of length. This "inexpressible diameter of five (squared)" means V50, in contrast to the pipij Sto^crpo^, the ** expressible diameter " of the same square, by which is meant the approxi- mation n/so-i, or 7. Thus for Euclid's predecessors \ '^•P would
apparently not have been rational but apprqro^, ** inexpressible," i.e. irrational.
I shall throughout my notes on this Book denote a rational straight line in Euclid's sense by p, and by p and o- when two different rational straight lines are required. Wherever then I use p or <r, it must be remembered that p, o- may have either of the forms a^ ^k . a, where a represents a units of length, a being either an integer or of the form m\n^ where w, n are both integers, and >( is an integer or of the form /«/« (where both w, n are integers) but not square. In other words, p, <r may have either of the forms a or JA^ where A represents A units of area and A is integral or of the form m\n^ where /», n are both integers. It has been the habit of writers to give a and ^a as the alternative forms of p, but I shall always use J A for the second in order to keep the dimensions right, because it must be borne in mind throughout that p is an irrational straight line.
As Euclid extends the signification of rational (fivfro^, literally expressible\ so he limits the scope of the term oXoyo? (literally having no ratio) as applied to straight lines. That this limitation was started by himself may perhaps be inferred from the form of words " let straight lines incommensurable with it be called irrational." Irrational straight lines then are with Euclid straight lines commensurable neither in length nor in square with the assumed rational straight line. Jk . a where k is not square is not irrational; i/k.a\s irrational, and so (as we shall see later on) is {»Jh± JX)a.
Definition 4.
Kai TO fi€tf awo rijs irportSuoTi^ cv^cias rerpdyiavov prjrov, koI ra rovrt^ ovfifi^Tpa ptjrdf ra 8c tovtw davfAfxerpa d\oya koXcio-^oi, ical at BvvdfJi^vat. avra aXoyoij ci fJL€v Trrpdywva cii/, avrai at TrXcvpa^ ci 3c crcpa ripa €v$vypafjLfi4i, at Ura avroic rcrpayoiva avaypa^ovaai.
As applied to areasj the terms rational and irrational have, on the other hand, the same sense with Euclid as we should attach to them. According to Euclid, if p is a rational straight line in his sense, p* is rational and any area commensurable with it, i.e. of the form kp^ (where k is an integer^ or of the form w/«, where m, n are integers), is rational ; but any area of the form ^k . p* is irrational, Euclid's rational area thus contains A units of area^ where A is an integer or of the form w/«, where m, n are integers; and his irrational area is of the form ^k.A, His irrational area is then connected with his irrational straight line by making the latter the square root of the
X. DEF. 4] NOTES ON DEFINITIONS 3, 4 13
former. This would give us for the irrational straight line !^k . JA^ which of course includes ijk . a.
at Swoficrai avra are the Straight lines the squares on which are equal to the areas, in accordance with the regular meaning of hvmirBau It is scarcely possible, in a book written in geometrical language, to translate hwaixivri as the square root (of an area) and Swao^ac as to be the square root (of an area)^ although I can use the term " square root " when in my notes I am using an algebraical expression to represent an area ; I shall therefore hereafter use the word "side" for Wa^cvi^ and "to be the side of" for SwVcur^ai, so that "side" will in such expressions be a short way of expressing the "side of a square equal to {an area)y In this particular passage it is not quite practi- cable to use the words " side of" or " straight line the square on which is equal to," for these expressions occur just afterwards for two alternatives which the word huvafkhrq covers. I have therefore exceptionally translated " the straight lines which produce them " (i.e. if squares are described upon them as sides).
al ura avroi^ rerpayiaya dyaypatf^owrai, literally " the (straight lines) which describe squares equal to them ' : a peculiar use of the active of dvaypd<fi€iv, the meaning being of course " the straight lines on which are described the squares " which are equal to the rectilineal figures.
BOOK X. PROPOSITIONS.
Proposition i.
Two unequal magnitudes being set out, if from the greater there be subtracted a magnitude greater than its half and from that which is left a magnitude greater than its half and if this process be repeated continually^ there will be left some magnitude which will be less than the lesser magnitude set out.
Let ABy C be two unequal magnitudes of which AB is the greater:
I say that, if from AB there be ^ — ' — ' »
subtracted a magnitude greater d 4^ + e
than its half, and from that which
is left a magnitude greater than its half, and if this process be repeated continually, there will be left some magnitude which will be less than the magnitude C,
For C if multiplied will -sometime be greater than AB.
[of. V. Def 4]
Let it be multiplied, and let DE be a multiple of C, and greater than AB ;
let DE be divided into the parts DF, FG, GE equal to C, from AB let there be subtracted BH greater than its half, and, from AH, HK greater than its half, and let this process be repeated continually until the divisions in AB are equal in multitude with the divisions in DE.
Let, then, AK, KHy HB be divisions which are equal in multitude with DF, FG, GE.
Now, since DE is greater than AB, and from DE there has been subtracted EG less than its half,
and, from AB, BH greater than its half, therefore the remainder GD is greater than the remainder HA.
X. i] PROPOSITION I 15
And, since GD is greater than HA, and there has been subtracted, from GD, the half GF, and, from HA, Z^A" greater than its half, therefore the remainder DF\s greater than the remainder AK.
But DF is equal to C; therefore C is also greater than AK.
Therefore AK is less than C.
Therefore there is left of the magnitude AB the magnitude AK which is less than the lesser magnitude set out, namely C
Q. E. D.
And the theorem can be similarly proved even if the parts subtracted be halves.
This proposition will be remembered because it is the lemma required in Euclid's proof of xii. 2 to the effect that circles are to one another as the squares on their diameters. Some writers appear to be under the impression that XII. 2 and the other propositions in Book xii. in which the method of exhaustion is used are the only places where Euclid makes use of x. i ; and it is commonly remarked that x. i might just as well have been deferred till the beginning of Book xii. Even Cantor {Gesch. d. Math, i,, p. 269) remarks that " Euclid draws no inference from it [x. i], not even that which we should more than anything else expect, namely that, if two magnitudes are incom- mensurable, we can always form a magnitude commensurable with the first which shall differ from the second magnitude by as little as we please." But, so far from making no use of x. i before xii. 2, Euclid actually uses it in the very next proposition, x. 2. This being so, as the next note will show, it follows that, since x. 2 gives the criterion for the incommensurability of two magnitudes (a very necessary preliminary to the study of incommensurables), x. I comes exactly where it should be.
Euclid uses x. i to prove not only xii. 2 but xii. 5 (that pyramids with the same height and triangular bases are to one another as their bases), by means of which he proves (xii. 7 and Por.) that any pyramid is a third part of the prism which has the same base and equal height, and xii. 10 (that any cone is a third part of the cylinder which has the same base and equal height), besides other similar propositions. Now xii. 7 Por. and xii. 10 are theorems specifically attributed to Eudoxus by Archimedes {On the Sphere and Cylinder, Preface), who says in another place {Quadrature of the Parabola, Preface) that the first of the two, and the theorem that circles are to one another as the squares on their diameters, were proved by means of a certain lemma which he states as follows : ''Of unequal lines, unequal surfaces, or unequal solids, the greater exceeds the less by such a magnitude as is capable, if added [continually] to itself, of exceeding any magnitude of those which are comparable with one another," i.e. of magnitudes of the same kind as the original magnitudes. Archimedes also says {loc» at,) that the second of the two theorems which he attributes to Eudoxus (Eucl. xii. 10) was proved by means of ''a lemma similar to^ the aforesaid." The lemma stated thus by Archimedes is decidedly different from x. i, which, however, Archimedes himself uses several times, while he refers to the use of it
i6 BOOK X [x. I
in XII. 2 {On the ^here and Cylinder^ i. 6). As I have before suggested (Tht Works of Archimedes^ p. xlviii), the apparent difficulty caused by the mention of tiifo lemmas in connexion with the theorem of Eucl. xii. 2 may be explained by reference to the proof of x. i. Euclid there takes the lesser magnitude and says that it is possible, by multiplying it, to make it some time exceed the greater, and this statement he clearly bases on the 4th definition of Book v., to the effect that "magnitudes are said to bear a ratio to one another which can, if multiplied, exceed one another." Since then the smaller magnitude in x. i may be regarded as the difference between some two unequal magnitudes, it is clear that the lemma stated by Archimedes is in substance used to prove the lemma in x. i, which appears to play so much larger a part in the investigations of quadrature and cubature which have come down to us.
Besides being employed in Eucl. x. i, the "Axiom of Archimedes" appears in Aristotle, who also practically quotes the result of x. i itself, l^hus he says. Physics viii. 10, 266 b 2, " By continually adding to a finite (magnitude) I shall exceed any definite (magnitude), and similarly by continually subtract- ing from it I shall arrive at something less than it," and ibid. iii. 7, 207 b 10 " For bisections of a magnitude are endless." It is thus somewhat misleading to use the term "Archimedes' Axiom" for the "lemma" quoted by him, since he makes no claim to be the discoverer of it, and it was obviously much earlier.
Stolz (quoted by G. Vitali in Questioni riguardanti la geometria elemetdare^ pp. 91 — 2) showed how to prove the so-called Axiom or Postulate of Archimedes by means of the Postulate of Dedekind, thus. Suppose the two magnitudes to be straight lines. It is required to prove that, given two straight lines ^ there always exists a multiple of the smaller which is greater tlian the other.
Let the straight lines be so placed that they have a common extremity and the smaller lies along the other on the same side of the common extremity.
\i AC ht the greater and AB the smaller, we have to prove that there exists an integral number n such that n . AB > AC,
Suppose that this is not true but that there are some points, like B^ not coincident with the extremity A^ and such that, n being any integer however great, n. AB <AC', and we have to prove that this assumption leads to an absurdity.
H .M \L^
A X Y B C
The points oi AC may be regarded as distributed into two "parts," namely (i) points -^for which there exists no integer n such that n . AI£> AC^ (2) points K for which an integer n does exist such that n . AK> AC,
This division into parts satisfies the conditions for the application of Dedekind's Postulate, and therefore there exists a point M such that the points of AM belong to the first part and those of MC to the second part
Take now a point Kon MC such that MY< AM, The middle point (X) oi AY will fall between A and M and will therefore belong to the first part ; but, since there exists an integer n such that n,AY>ACi it follows that 2n.AX> AC: which is contrary to the hypothesis.
X. 2] PROPOSITIONS I, 2 17
Proposition 2.
If^ when the less of two unequal magnitudes is continually subtracted in turn from the greater, that which is left never measures the one before it, the magnitudes will be incom- mensurable.
For, there being two unequal magnitudes AB, CD, and AB being the less, when the less is continually subtracted in turn from the greater, let that which is left over never measure the one before it ; I say that the magnitudes AB, CD are incommensurable.
_E_ A-S B
0 + D
For, if they are commensurable, some magnitude will measure them.
Let a magnitude measure them, if possible, and let it be E; let AB, measuring FD, leave CF less than itself, let CF measuring BG, leave AG less than itself, and let this process be repeated continually, until there is left some magnitude which is less than E.
Suppose this done, and let there be left AG less than E.
Then, since E measures AB, while AB measures DF, therefore E will also measure FD.
But it measures the whole CD also ; therefore it will also measure the remainder CF.
But CF measures BG ; therefore E also measures BG.
But it measures the whole AB also ; therefore it will also measure the remainder A G, the greater the less : which is impossible.
Therefore no magnitude will measure the magnitudes AB, CD,
therefore the magnitudes AB, CD are incommensurable.
[x. Def. i] Therefore etc.
H. E. III. 2
1 8 BOOK X [x. 2
This proposition states the test for incommensurable magnitudes, founded on the usual operation for finding the greatest common measure. The sign of the incommensurability of two magnitudes is that this operation never comes to an end, while the successive remainders become smaller and smaller until they are less than any assigned magnitude.
Observe that Euclid says " let this process be repeated continually until there is left some magnitude which is less than E^ Here he evidently assumes that the process mil some time produce a remainder less than any assigned magnitude E, Now this is by no means self-evident, and yet Heiberg (though so careful to supply references) and Lorenz do not refer to the basis of the assumption, which is in reality x. i, as Billingsley and Williamson were shrewd enough to see. The fact is that, if we set off a smaller magnitude once or oftener along a greater which it does not exactly measure, until the remainder is less than the smaller magnitude, we take away from the greater more than its half. Thus, in the figure, FD is more than the half of CDy and BG more than tiie half of AB, If we continued the process, AG marked off along CF as many times as possible would cut off more than its half; next, more than half AG would be cut off, and so on. Hence along CZ?, AB alternately the process would cut off more than half, then more than half the remainder and so on, so that on both lines we should ultimately arrive at a remainder less than any assigned length.
The method of finding the greatest common measure exhibited in this proposition and the next is of course again the same as that which we use and which may be shown thus :
b)a(p
Pi c)b{q
d)c{r rd
The proof too is the same as ours, taking just the same form, as shown in the notes to the similar propositions vii. i, 2 above. In the present case the hypothesis is that the process never stops, and it is required to prove that a, b cannot in that case have any common measure, as/ For suppose that/ is a common measure, and suppose the process to be continued until the remainder ^, say, is less than/
Then, since /measures a, by it measures a -pby or c.
Since / measures ^, r, it measures b-qcy or //; and, since/ measures r, </, it measures c — rdyOxe\ which is impossible, since e </
Euclid assumes as axiomatic that, if/ measures a, ^, it measures ma ± nb.
In practice, of course, it is often unnecessary to carry the process far in order to see that it will never stop, and consequently that the magnitudes are incommensurable. A good instance is pointed out by Allman (Greek Geometry from Thales to Euclid, pp. 42, 137—8). Euclid proves in xiii. 5 that, if AB be cut in extreme and mean ratio at 6*, and if
DA equal to ^C be added, then DB is also cut D A c b
in extreme and mean ratio at A. This is indeed
obvious from the proof of 11. 11. It follows conversely that, if BD is cut into extreme and mean ratio at Ay and AC, equal to the lesser segment ADy be subtracted from the greater ABy AB is similarly divided at C. We can then
X.2] PROPOSITION 2 19
mark off from ^ C* a portion equal to CB^ and A C will then be similarly divided, and so on. Now the greater segment in a line thus divided i^ greater than half the line, but it follows from xiii. 3 that it is less than twice the lesser s^ment, i.e. the lesser segment can never be marked off more than once from the greater. Our process of marking off the lesser segment from the greater continually is thus exactly that of finding the greatest common measure. If, therefore, the segments were commensurable, the process would stop. But it clearly does not ; therefore the segments are incommensurable.
Allman expresses the opinion that it was rather in connexion with the line cut in extreme and mean ratio than with reference to the diagonal and side of a square that Pythagoras discovered incommensurable magnitudes. But the evidence seems to put it beyond doubt that the Pythagoreans did discover the incommensurability of J2 and devoted much attention to this particular case. The view of Allman does not therefore commend itself to me, though it is likely enough that the Pythagoreans were aware of the incommensura- bility of the segments of a line cut in extreme and mean ratio. At all events the Pythagoreans could hardly have carried their investigations into the in- commensurability of the segments of this line very far, since Theaetetus is said to have made the first classification of irrationals, and to him is also, with reasonable probability, attributed the substance of the first part of Eucl. XIII., in the sixth proposition of which occurs the proof that the segments of a rational straight line cut into extreme and mean ratio are apotomes.
Again, the incommensurability of ^2 can be proved by a method practically equivalent to that of x. 2, and without carrying the process very far. This method is given in ChrystaFs Text- book of Algebra (i. p. 270). Let d^ a be the diagonal and side respectively of a square A BCD, Mark off ^7?* along AC equal to a. Draw FE at right angles to -^C meeting BC \nE.
It is easily proved that
BE = EF= FC, Qc'
CF=AC-AB = d-a (i).
CE^CB- CF^a-^id-a)
= 2a — d (2).
Suppose, if possible, that dy a are commensurable. If d^ a are both commensurably expressible in terms of any finite unit, each must be an integral multiple of a certain finite unit
But from (i) it follows that CF^ and from (2) it follows that CE^ is an integral multiple of the same unit
And CFy CE are the side and diagonal of a square CFEG, the side of which is less than half the side of the original square. U ai^di are the side and diagonal of this square,
ai = d-a ^ di = 2a-d { '
Similarly we can form a square with side a^ and diagonal d^ which are less than half ^1, di respectively, and a^, d^ must be integral multiples of the same unit, where
di = 2^1 - d, ;
2 — 2
20 BOOK X [x. 2, 3
and this process may be continued indefinitely until (x. i) we have a square as small as we please, the side and diagonal of which are int^ral multiples of a finite unit : which is absurd.
Therefore a, d are incommensurable.
It will be observed that this method is the opposite of that shown in the Pythagorean series of side- and diagonal-numbers, the squares being successively smaller instead of larger.
Proposition 3.
Given two commensurable magnitudes y to find their greatest common measure.
Let the two given commensurable magnitudes be AB, CD of which AB is the less ;
thus it is required to find the greatest common measure of AB, CD.
Now the magnitude AB either measures CD or it does not.
If then it measures it — and it measures itself also — AB is a common measure of AB, CD.
And it is manifest that it is also the greatest ;
for a greater magnitude than the magnitude AB will not measure AB.
--^ a4 B
C g O
Next, let AB not measure CD.
Then, if the less be continually subtracted in turn from the greater, that which is left over will sometime measure the one before it, because AB, CD are not incommensurable ;
[cf. X. 2] let AB, measuring ED, leave EC less than itself,
let EC, measuring FB, leave AF less than itself,
and let AF measure CE.
Since, then, AF measures CE, while CE measures FB, therefore AF will also measure FB.
But it measures itself also ; therefore AF will also measure the whole AB.
X. 3] PROPOSITIONS 2, 3 21
But AB measures DE ; therefore AF will also measure ED.
But it measures CE also ; therefore it also measures the whole CD.
Therefore AF is a common measure of A By CD,
I say next that it is also the greatest.
For, if not, there will be some magnitude greater than -^/^ which will measure ABy CD.
Let it be G.
Since then G measures AB, while AB measures ED, therefore G will also measure ED. •
But it measures the whole CD also ; therefore G will also measure the remainder CE.
But CE measures FB ; therefore G will also measure FB.
But it measures the whole AB also, and it will therefore measure the remainder AF, the greater the less : which is impossible.
Therefore no magnitude greater than AF will measure AB, CD', therefore AF is the greatest common measure of AB, CD.
Therefore the greatest common measure of the two given commensurable magnitudes AB, CD has been found.
Q. E. D.
PoRiSM. From this it is manifest that, if a magnitude measure two magnitudes, it will also measure their greatest common measure.
This proposition for two commensurable magnitudes is, mutatis mutandis^ exactly the same as vii. 2 for numbers. We have the process
b)a{p
c)b(3
d)c(r rd
where e is equal to rd and therefore there is no remainder,
22 BOOK X [x. 3, 4
It is ^hen proved that ^ is a common measure of a^d; and next, by a reductio ad absurdum^ that it is the greatest common measure, since any common measure must measure d^ and no magnitude greater than d can measure d. The redtutio ad absurdum is of course one of form only.
The Porism corresponds exactly to the Porism to vii. 2.
The process of finding the greatest common measure is probably given in this Book) not only for the sake of completeness, but because in x. 5 a common measure of two magnitudes A^ B is assumed and used, and therefore it is important to show that such a measure can be found if not already known.
Proposition 4.
Given three commensurable magnitudes, to find their greatest common measure.
Let A, By C be the three given commensurable magnitudes; thus it is required to find the greatest common measure of A, By C. A
Let the greatest common measure b
of the two magnitudes A, B be taken, c
and let it be ^ ; [x. 3] ^ ^ p
then D either measures C, or does not measure it.
First, let it measure it.
Since then D measures C, while it also measures A, B, therefore Z? is a common measure of -^, B, C.
And it is manifest that it is also the greatest ;
for a greater magnitude than the magnitude D does not measure A, B,
Next, let Z? not measure C.
I say first that C, D are commensurable.
For, since A, B, C are commensurable,
some magnitude will measure them,
and this will of course measure A, B also ;
so that it will also measure the greatest common measure of A, By namely D. [x. 3, Por.]
But it also measures C ;
so that the said magnitude will measure C, D ;
therefore C, D are commensurable.
X. 4] PROPOSITIONS 3, 4 23
Now let their greatest common measure be taken, and let it be E. [x. 3]
Since then E measures D, while D measures Ay B, therefore E will also measure A, B.
But it measures C also ; therefore E measures A, B, C; therefore E is 3, common measure of -^, B, C.
I say next that it is also the greatest. For, if possible, let there be some magnitude /^greater than E, and let it measure A, B, C.
Now, since E measures A, B, C,
it will also measure A, B,
and will measure the greatest common measure of A, B.
[x. 3, Por.] But the greatest common measure o( A, B is D ;
therefore E measures D.
But it measures C also ;
therefore E measures C, Z? ;
therefore E will also measure the greatest common measure of C D. [x. 3, Por.]
But that is E ;
therefore E will measure E, the greater the less :
which is impossible.
Therefore no magnitude greater than the magnitude E will measure A, B, C\
therefore E is the greatest common measure of ^, By C if D do not measure C,
and, if it measure it, Z? is itself the greatest common measure.
Therefore the greatest common measure of the three given commensurable magnitudes has been found.
PoRiSM. From this it is manifest that, if a magnitude measure three magnitudes, it will also measure thejr greatest common measure.
Similarly too, with more magnitudes, the greatest common measure can be found, and the porism can be extended.
Q. E. D.
24 BOOK X [x. 4, 5
This proposition again corresponds exactly to vii. 3 for numbers. As there Euclid thinks it necessary to prove that, a, by c not being prime to one another, d and ^ are also not prime to one another, so here he thinks it necessary to prove that dy c are commensurable, as they must be since any common measure of a, b must be a measure of their greatest common measure d (x. 3, Por.).
The argument in the proof that ^, the greatest common measure of dy Cy is the greatest common measure of a, by r, is the same as that in vii. 3 and x. 3.
The Porism contains the extension of the process to the case of four or more magnitudes, corresponding to Heron's remark with regard to the similar extension of vii. 3 to the case of four or more numbers.
Proposition 5.
Commensurable magnitudes have to one another the ratio which a number has to a number.
Let Ay B be commensurable magnitudes ;
I say that A has to B the ratio which a number has to a number.
For, since Ay B zxt, commensurable, some magnitude will measure them.
Let it measure them, and let it be C
And, as many times as C measures Ay so many units let there be in D ;
and, as many times as C measures By so many units let there be in ^.
Since then C measures A according to the units in /?,
while the unit also measures D according to the units in it,
therefore the unit measures the number D the same number of times as the magnitude C measures A ;
therefore, as C is to Ay so is the unit to D ; [vii. Def. 20]
therefore, inversely, as A is to C, so is D to the unit.
[cf. v. 7, Por.] Again, since C measures B according to the units in Ey while the unit also measures E according to the units in it,
X. 5] PROPOSITIONS 4, S 25
therefore the unit measures E the same number of times as C measures B ;
therefore, as C is to B^ so is the unit to E. But it was also proved that,
as y4 is to C, so is D to the unit ; therefore, ex aeguali,
as y4 is to Bj so is the number D to E. [v. 22]
Therefore the commensurable magnitudes A, B have to one another the ratio which the number D has to the number E.
Q. E, D.
The argument is as follows. \i a, b he commensurable magnitudes, they have some common measure r, and
|
a = m€y |
|
|
d = nc, |
|
|
where Mj n are integers. |
|
|
It follows that |
c:a= 1 :m . |
|
or, inversely, |
a \c = m\ i; |
|
and also that |
<: : ^ = I : «, |
|
so that, ex aequali^ |
a:d = m:n. |
•(>).
It will be observed that, in stating the proportion (i), Euclid is merely expressing the fact that a is the same multiple of c that »i is of i. In other words, he rests the statement on the definition of proportion in vii. Def. 20. This, however, is applicable only to four numbers, and c, a are not numbers but magnitudes. Hence the statement of the proportion is not legitimate unless it is proved that it is true in the sense of v. Def. 5 with regard to magnitudes in general, the numbers i, m being magnittides. Similarly with regard to the other proportions in the proposition.
There is, therefore, a hiatus. Euclid ought to have proved that magnitudes which are proportional in the sense of vii. Def. 20 are also proportional in the sense of v. Def. 5, or that the proportion of numbers is included in the proportion of magnitudes as a particular case. Simson has proved this in his Proposition C inserted in Book v. (see Vol. 11. pp. 126—8). The portion of that proposition which is required here is the proof that, if a-mb \
c = mdr then a:b = c:dy in the sense of v. Def. 5.
Take any equimultiples pa, pc of a, c and any equimultiples qb, qd of by d.
Now pa =pmb \
pc=pmd) '
But, according as pmb > = < qb,pmd> = < qd. Therefore, according 2^pa > = <qb,pa> = < qd.
And pa, pc are any equimultiples of a, c, and qb, qd any equimultiples of b, d.
Therefore a\b = c\d. [v. Def. 5.]
26 BOOK X [x. 6
Proposition 6.
If two magnitudes have to one another the ratio which a number has to a number^ the magnitudes will be commensurable.
For let the two magnitudes A, B have to one another the ratio which the number D has to the number E ; s I say that the magnitudes A, B are commensurable.
A ' \ B C
O
E F
For let A be divided into as many equal parts as there are units in /?,
and let C be equal to one of them ;
and let F be made up of as many magnitudes equal to C as lo there are units in E.
Since then there are in ^ as many magnitudes equal to C
as there are units in D,
whatever part the unit is of /?, the same part is C of -^ also ;
therefore, as C is to ^, so is the unit to D. [vii. Def. 20]
15 But the unit measures the number D ;
therefore C also measures A.
And since, as C is to Ay so is the unit to /?,
therefore, inversely, as -^ is to C, so is the number D to the
unit. [of. V. 7, Por.]
20 Again, since there are in -F as many magnitudes equal to C as there are units in E,
therefore, as C is to /% so is the unit to E. [vii. Def. 20]
But it was also proved that,
as A is to C, so is D to the unit ; 25 therefore, ex aequali, as y? is to /% so is D to E. [v. 22]
But, as Z? is to E, so is -^ to ^ ; therefore also, as ^ is to B, so is it to /^also. [v. n]
Therefore A has the same ratio to each of the magnitudes By F\ 30 therefore B is equal to F, [v. 9]
But C measures Fy therefore it measures B also.
Further it measures A also ; therefore C measures Ay B,
X. 6] PROPOSITION 6 27
35 Therefore A is commensurable with B, Therefore etc.
PoRiSM. From this it is manifest that, if there be two numbers, as D, E, and a straight line, as -^4, it is possible to make a straight line \F'\ such that the given straight line is to • 40 it as the number D is to the number E.
And, if a mean proportional be also taken between A, /% as^,
as ^ is to F, so will the square on A be to the square on B,
that is, as the first is to the- third, so is the figure on the first
45 to that which is similar and similarly described on the second.
[vi. 19, Por.]
But, as A is to /% so is the number D to the number E ; therefore it has been contrived that, as the number D is to the number E, so also is the figure on the straight line A to the figure on the straight line B. q. e. d.
15. But the unit measures the number D ; therefore C also measures A. These words are redundant, though they are apparently found in all the Mss.
The same link to connect the proportion of numbers with the proportion of magnitudes as was necessary in the last proposition is necessary here. This being premised, the argument is as follows.
Suppose a\b = m\n^
where m, n are (integral) numbers.
Divide a into m parts, each equal to ^, say, so that a^mc.
Now take d such that d= nc.
Therefore we have a\c = m\i^
and c:d- I :n,
so that, €X aequaliy a : d= m : n
= a : ^, by hypothesis.
Therefore h = d-nc^ so that c measures b n times, and a^ b are commensurable.
The Porism is often used in the later propositions. It follows (i) that, if a be a given straight line, and m, n any numbers, a straight line x can be found such that
a\x — m\n, (2) We can find a straight line j^ such that
a^\j^ = m :«. For we have only to take 7, a mean proportional between a and x^ as
Effeviously found, in which case a, y^ x are in continued proportion and V. Def. 9]
a*iy-a\x
28 BOOK X [x. 7—9
Proposition 7.
Incommensurable magnitudes have not to one another the ratio which a number has to a number.
Let A, B be incommensurable magnitudes ; I say that A has not to B the ratio which a number has to a number.
For, if A has to B the ratio which a number has to a number, A will be commensurable with B, [x. 6]
But it is not ; therefore A has not to B the ratio which a 5
number has to a number.
Therefore etc.
Proposition 8.
If two magnitudes have not to one another the ratio which a number has to a number^ the magnitudes will be incom- mensurable.
For let the two magnitudes A, B not have to one another the ratio which a number has to a number ;
I say that the magnitudes A, B are incom- — ~
mensurable.
For, if they are commensurable, A will have to B the ratio which a number has to a number. [x. 5]
But it has not ; therefore the magnitudes A^ B are incommensurable.
Therefore etc.
Proposition 9.
The squares on straight lines commensurable in length have to one another the ratio which a square number has to a square number ; and squares which have to one another the rcUio which a square number has to a square number will also ha^e their sides commensurable in length. But the squares on straight lines incommensurable in length have not to one another the ratio which a square number has to a square number ; and squares which have not to one another the ratio which a square number has to a square number will not have their sides commensurable in length either.
X. 9] PROPOSITIONS 7—9 29
For let A J B be commensurable in length ; I say that the square on ^4 ^
has to the square on B the ^
ratio which a square number ~^
has to a square number.
For, since A is commensurable in length with B, therefore A has to B the ratio which a number has to a number. [x. 5]
Let it have to it the ratio which C has to D.
Since then, as A is to B, so is C to /?, while the ratio of the square on A to the square on B is duplicate of the ratio of A to B,
for similar figures are in the duplicate ratio of their corre- sponding sides ; [vi. 20, Por.]
and the ratio of the square on C to the square on D is duplicate of the ratio of C to D,
for between two square numbers there is one mean proportional number, and the square number has to the square number the ratio duplicate of that which the side has to the side ; [viii. n] therefore also, as the square on A is to the square on B, so is the square on C to the square on /?.
Next, as the square on A is to the square on B, so let the square on C be to the square on Z? ; I say that A is commensurable in length with B.
For since, as the square on A is to the square on By so is the square on C to the square on Z?,
while the ratio of the square on A to the square on B is duplicate of the ratio of A to B,
and the ratio of the square on C to the square on D is duplicate of the ratio of C to Dy therefore also, as A is to By so is C to D.
Therefore A has to B the ratio which the number C has to the number D ; therefore A is commens.urable in length with B. [x. 6]
Next, let A be incommensurable in length with B \ I say that the square on A has not to the square on B the ratio which a square number has to a square number.
For, if the square on A has to the square on B the ratio
30 BOOK X [x. 9
which a square number has to a square number, A will be commensurable with B.
But it is not ; therefore the square on A has not to the square on B the ratio which a square number has to a square number.
Again, let the square on A not have to the square on B the ratio which a square number has to a square number ; I say that A is incommensurable in length with B.
For, if A is commensurable with B, the square on A will have to the square on B the ratio which a square number has to a square number.
But it has not ; therefore A is not commensurable in length with B.
Therefore etc.
PoRiSM. And it is manifest from what has been proved that straight lines commensurable in length are always com- mensurable in square also, but those commensurable in square are not always commensurable in length also.
[Lemma. It has been proved in the arithmetical books that similar plane numbers have to one another the ratio which a square number has to a square number, [vni. 26]
and that, if two numbers have to one another the ratio which a square number has to a square number, they are similar plane numbers. [Converse of viii. 26]
And it is manifest from these propositions that numbers which are not similar plane numbers, that is, those which have not their sides proportional, have not to one another the ratio which a square number has to a square number.
For, if they have, they will be similar plane numbers : which is contrary to the hypothesis.
Therefore numbers which are not similar plane numbers have not to one another the ratio which a square number has to a square number.]
A scholium to this proposition (Schol. x. No. 62) says categorically that the theorem proved in it was the discovery of Theaetetus.
If a, ^ be straight lines, and
a ib^mifiy where w, n are numbers,
then <i»:^ = «•:««;
and conversely.
X. 9, lo] PROPOSITIONS 9, 10 31
This inference, which looks so easy when thus symbolically expressed, was by no means so easy for Euclid owing to the fact that a, b are straight lines, and «r, n numbers. He has to pwiss from a : ^ to a* : ^ by means of vi. 20, Por. through the duplicate ratio; the square on a is to the square on b in the duplicate ratio of the corresponding sides a, b. On the other hand, m, n being numbers^ it is viii. 1 1 which has to be used to show that n^ : t^ is the ratio duplicate oim\n.
Then, in order to establish his result, Euclid assumes that, ifiwo ratios are equals the ratios which are their duplicates are also equal. This is nowhere proved in Euclid, but it is an easy inference from v. 22, as shown in my note on VI. 22.
The converse has to be established in the same careful way, and Euclid assumes that ratios the duplicates of which are equal are themselves equal. This is much more troublesome to prove than the converse; for proofs I refer to the same note on vi. 22.
The second part of the theorem, deduced by reductio ad absurdum from the first, requires no remark.
In the Greek text there is an addition to the Porism which Heiberg brackets as superfluous and not in Euclid's manner. It consists (i) of a sort of proof, or rather explanation, of the Porism and (2) of a statement and explanation to the effect that straight lines incommensurable in length are not necessarily incommensurable in square also, and that straight lines incommensurable in square are, on the other hand, always incommensurable in length also.
The Lemma gives expressions for two numbers which have to one another the ratio of a square number to a square number. Similar plane numbers arc of the form pm . pn and qm . qn^ or mnf^ and mn^^ the ratio of which is of course the ratio of /* to f.
The converse theorem that, if two numbers have to one another the ratio of a square number to a square number, the numbers are similar plane numbers is not, as a matter of fact, proved in the arithmetical Books. It is the converse of viii. 26 and is used in ix. 10. Heron gave it (see. note on VIII. 27 above).
Heiberg however gives strong reason for supposing the Lemma to be an interpolation. It has reference to the next proposition, x. 10, and, as we shall see, there are so many objections to x. 10 that it can hardly be accepted as genuine. Moreover there is no reason why, in the Lemma itself, numbers which are not similar plane numbers should be brought in as they are.
[Proposition 10.
To find two straight lines incommensurable^ the one in length only, and the other in square also, with an assigned straight line.
Let A be the assigned straight line ; thus it is required to find two straight lines incommensurable, the one in length only, and the other in square also, with A.
Let two numbers By C be set out which have not to one
32 BOOK X [x. lo
another the ratio which a square number has to a square number, that is, which are not similar plane
numbers ; A
and let it be contrived that, o
as B is to C, so is the square on A to the square on D
c
— for we have learnt how to do this —
[x. 6, Por.]
therefore the square on A is commensurable with the square on /?. [x. 6]
And, since B has not to C the ratio which a square number has to a square number,
therefore neither has the square on A to the square on D the ratio which a square number has to a square number ; therefore A is incommensurable in length with D. [x. 9]
Let E be taken a mean proportional between Ay D \
therefore, as A is to /?, so is the square on A to the square on E. [v. Def. 9]
But A is incommensurable in length with D ; therefore the square on A is also incommensurable with the square on jfi"; [*• 11]
therefore A is incommensurable in square with E,
Therefore two straight lines /?, E have been found in- commensurable, D in length only, and E in square and of course in length also, with the assigned straight line ^.]
It would appear as though this proposition was intended to supply a justification for the statement in x. Def. 3 that it is proved that there are an infinite number of straight lines {a) incommensurable in length only, or commensurable in square only, and (b) incommensurable in square, with any given straight line.
But in truth the proposition could well be dispensed with; and the positive objections to its genuineness are considerable.
In the first place, it depends on the following proposition, x. 1 1 ; for the last step concludes that, since
c^ \f = a \Xy
and a, x are incommensurable in length, therefore a^, ^ are incommensurable. But Euclid never commits the irregularity of proving a theorem by means of a later one. Gregory sought to get over the difficulty by putting x. 10 after X. 1 1 ; but of course, if the order were so inverted, the Lemma would still be in the wrong place.
Further, the expression ifjLaOofi€v yap, "for we have learnt (how to do this)," is not in Euclid's manner and betrays the hand of a learner (though the same
"^
X. lo, ii] PROPOSITIONS lo, ii 33
expression is found in the Sectio Canonis of Euclid, where the reference is to the Elements).
Lastly the manuscript P has the number 10, in the first hand, at the top of X. II, from which it may perhaps be concluded that x. 10 had at first no number.
It seems best therefore to reject as spurious both the Lemma and x. 10.
The argument of x. 10 is simple. If a be a given straight line and /», n numbers which have not to one another the ratio of square to square, take x such that
a^ \ x^^tn\ «, [x. 6, Por.]
whence a, x are incommensurable in length. [x. 9]
Then take y a mean proportional between a, at, whence
a} \y^-a \ X [v. Def. 9]
[- sIm : >/«], and X is incommensurable in length only, while y is incommensurable in square as well as in length, with a.
Proposition 11.
If four magnitudes be proportionaly and the first be com- mensurable with the second, the third will also be commensurable with the fourth ; and, if the first be incommensurable with the second, the third will also be incommensurable with the fourth.
Let A, B, C, D he, four magnitudes in proportion, so that, as A is to B, so is C
to D, A B
and let A be commensurable ^ d
with B ;
I say that C will also be commensurable with D.
For, since A is commensurable with B, therefore A has to B the ratio which a number has to a number. [x. 5]
And, as A is to B, so is C to Z? ; therefore C also has to D the ratio which a number has to a number ; therefore C is commensurable with D. [x. 6]
Next, let A be incommensurable with B ; I say that C will also be incommensurable with D,
For, since A is incommensurable with B, therefore A has not to B the ratio which a number has to a number. [x. 7]
H. E. 111. 3
34 BOOK X [x. II, 12
And, as A is to B, so is C to Z> ; therefore neither has C to D the ratio which a number has to a number ; therefore C is incommensurable with D. [x. 8]
Therefore etc.
I shall henceforth, for the sake of brevity, use symbols for the terms "commensurable (with)" and "incommensurable (with)*' according to the varieties described in x. Deff. i — 4. The symbols are taken from Lorenz and seem convenient.
Commensurable and commensurable with^ in relation to areas, and com- mensurable in length and commensurahle in length ivith^ in relation to straight lines, will be denoted by ^,
Commensurable in square only or commensurable in square only with (terms applicable only to straight lines) will be denoted by r^ .
Incommensurable {with)^ of areas, and incommensurable {with\ of straight lines will be denoted by ^^ .
Ificommensurable in square (with) (a term applicable to straight lines only) will be denoted by w-.
Suppose a, ^, r, ^ to be four magnitudes such that
a : b = c : d.
Then (i), if a ^ ^, a : b = m : n, where ///, n are integers, [x. 5]
whence c : d-m : n,
and therefore c ^ d. [x. 6]
(2) U a ^ b, a : b=¥m : n, [x. 7]
so that c : d^m : «,
whence c s^ d, [x. 8]
Proposition 12.
MagnitucUs commensurable with the same magnitude are commensurable with one another also.
For let each of the magnitudes A, B be commensurable with C; I say that A is also commensurable with B.
-D
— E H
-F K
-Q L
For, since A is commensurable with C, therefore A has to C the ratio which a number has to a number. [x. 5]
X. 12] PROPOSITIONS II, li 3^
Let it have the ratio which D has to E. Again, since C is commensurable with B,
therefore C has to B the ratio which a number has to a number. [x. 5]
Let it have the ratio which F has to G. And, given any number of ratios we please, namely the ratio which D has to E and that which F has to G,
let the numbers H, K, L be taken continuously in the given ratios ; [of. vm. 4]
so that, as Z? is to E, so is H to Ky
and, as /^ is to G, so is K to Z.
Since, then, as A is to C, so is D to E^ while, as D is to E, so is 1/ to A', therefore also, as A is to C, so is I/to K. [v. n]
Again, since, as C is to B, so is F to Gy while, as /^ is to G^, so is A" to Z, therefore also, as C is to By so is A" to L, [v. n]
But also, as A is to C, so\sHXjoK\
therefore, ex aequaliy as A is to ^ff, so is H to Z. [v. 22]
Therefore ^4 has to B the ratio which a number has to a number ;
therefore A is commensurable with B, [x. 6]
Therefore etc.
Q. E. D.
|
We have merely to numbers. |
go |
through the process of |
compounding two |
ratios in |
|
Suppose |
a, b each *^ c. |
|||
|
Therefore |
a \c=m \ny say, c\b = p\qy say. |
[X.5] |
||
|
Now |
tn :n = mp : npy |
|||
|
and |
p:q = np: nq. |
|||
|
Therefore |
a \c~mp \ npy c : b = np : nq, |
|||
|
whence, ex aequaliy |
a : b = mp : nq, |
|||
|
so that |
a "^ b. |
[x.6] |
3—2
36 BOOK X [x. 13, Lemma
Proposition 13.
If two magnitudes be commensurable, and the one of them be incommensurable with any magnitudey the remaining one will also be incommensurable with the same.
Let A, B be two commensurable magnitudes, and let one of them, A, be incommensurable with
any other magnitude C\ a
I say that the remaining one, B, will o^
also be incommensurable with C b
For, if B is commensurable with C, while A is also commensurable with By A is also commensurable with C [x. 12]
But it is also incommensurable with it : which is impossible.
Therefore B is not commensurable with C ; therefore it is incommensurable with it.
Therefore etc.
Lemma.
Given two unequal straight lines, to find by what square the square on the greater is greater than the square on the less.
Let AB, C be the given two unequal straight lines, and let AB be the greater of them ; thus it is required to find by what square the square on AB is greater than the square on C.
Let the semicircle ADB be de- scribed on AB,
and let AD\y^ fitted into it equal to C ; [iv. i]
let DB be joined.
It is then manifest that the angle ADB is right, [ni. 31]
and that the square on AB is greater than the square on AD, that is, C, by the square on DB. [i. 47]
Similarly also, if two straight lines be given, the straight line the square on which is equal to the sum of the squares on them is found in this manner.
Lemma, x. 14] PROPOSITIONS 13, 14 37
Let AD, DB be the given two straight lines, and let it be required to find the straight line the square on which is equal to the sum of the squares on them.
Let them be placed so as to contain a right angle, that formed by AD, DB ; and let AB be joined.
It is again manifest that the straight line the square on which is equal to the sum of the squares on AD, DB is AB,
[I. 47]
Q. E. D.
The lemma gives an obvious method of finding a straight line (c) equal to ^/a*T^, where a, b are given straight lines of which a is the greater.
Proposition 14.
If four straight lines be proportional, and the square on
the first be greater than the square on the second by the square
an a straight line commensurable with the first, the square on
the third will also be greater than the square on the fourth by
5 the square on a straight line commensurable with the third.
And, if the square on the first be greater than the square
on the second by the square on a straight line incommensurable
with the first, the square on the third will also be greater than
the square on the fourth by the square on a straight line in-
\o commensurable with the third.
Let A, B, C D be four straight lines in proportion, so
that, as .^ is to B, so is C to Z? ;
and let the square oA A be greater than
the square on B by the square on E, and islet the square on C be greater than the
square on D by the square on F\
I say that, if A is commensurable with E,
C is also commensurable with F,
and, if A is incommensurable with E, C is ao also incommensurable with F,
For since, as ^4 is to B, so is C to D,
therefore also, as the square on A is to the square on B, so is
the square on C to the square on D. [vi. 22]
But the squares on E, B are equal to the square on A, 25 and the squares on D, F are egual to the square on C.
A B CD
38 BOOK X [x. 14
Therefore, as the squares on E, B are to the square on By so are the squares on /?, F to the square on D ;
therefore, separando, as the square on E is to the square on B, so is the square on F to the square on D ; [v. 17]
30 therefore also, as E is to B, so is F to D \ [vi. 22]
therefore, inversely, as B is to E^ so is D to F.
But, as A is to B, so also is C to Z> ;
therefore, ex aequaliy as A is to E, so is C to -/^ [v. 22]
Therefore, if A is commensurable with E, C is also com- 35 mensurable with /%
and, if A is incommensurable with E, C is also incommen- surable with /^ [x. 11]
Therefore etc.
3i 5) 8, 10. Euclid speaks of the square on the first (third) being greater than the square on the second (fourth) by the square on a straight line commensurable (incommensurable) *' with itself (kavT^y^ and similarly in all like phrases throughout the Book. For clearness sake I substitute **the first," ** the third," or whatever it may be, for " Itself" in these cases.
Suppose a, ^, ^, ^ to be straight lines such that
a\b = c\d (i).
It follows [vi. 22] that a* : ^ = r* . d* (2).
In order to prove that, convertendoy
a':(a^-i^) = c^:{c^-d^)
Euclid has to use a somewhat roundabout method owing to the absence of a convertendo proposition in his Book v. (which omission Simson supplied by his Prop. E).
It follows from (2) that
whence, separando^ (a* - ^) : ^ = (^ ' - ^) : ^^ [v. 17]
and, inversely, ^ : («» - ^) = //» : (r« - d").
From this and {2\€x aequali^
««:(a«-^) = r*:(r«-^/«). [v. 22]
Hence a : Va* --li^^c : Jc* — </*. [vi. 22]
According therefore as a^or^ Jc^ - ^,
c*^ox^Jc^-d^, [x. 11]
If a '^ *Ja^ - ^*, we may put Jd^ -if^ = ka, where k is of the form mjn and »i, n are integers. And if Ja'-i^-ka^ it follows in this case that
X. is] propositions 14, 15 39
Proposition 15.
If two commensurable magnitudes be added together^ the whole will also be commensurable with ea4:h of th^ ; and, if the whole be commensurable with one of them, the original magnitudes will also be commensurable.
For let the two commensurable magnitudes AB, BC be added together ;
I say that the whole AC is also a 1 o
commensurable with each of the j
magnitudes AB, BC.
For, since AB, BC are commensurable, some magnitude will measure them.
Let it measure them, and let it be D.
Since then D measures AB, BC, it will also measure the whole AC
But it measures AB, BC also ; therefore D measures AB, BC, AC;
therefore ^C is commensurable with each of the magnitudes AB, BC. [x. Def. i]
Next, let AC be commensurable with AB; I say that AB, BC are also commensurable.
For, since AC, AB are commensurable, some magnitude will measure them.
Let it measure them, and let it be D.
Since then D measures CA, AB, it will also measure the remainder BC
But it measures AB also ;
therefore Z? will measure AB, BC ;
therefore AB, BC are commensurable. [x. Def. i]
Therefore etc.
(i) If a, d he any two commensurable magnitudes, they are of the form mc, nc, where r is a common measure of a, b and m, n some integers.
It follows that a + b = {m + n)c;
therefore {a + d>), being measured by c, is commensurable with both a and d.
(2) If a + ^ is commensurable with either a or d, say a, we may put a + i = fnc, a = nc, where ^ is a common measure of (a + b), a, and m, n are integers.
Subtracting, we have b = (m— n) c,
whence b ^ a.
40 BOOK X [x. i6
Proposition i6.
If two incommensurable magnitudes be added together, the whole will also be incommensurable with each of them : and, if the whole be incommensurable with one of them, the original magnitudes will also be incommensurable.
For let the two incommensurable magnitudes AB, BC be added together ;
I say that the whole AC is also incommensurable with each of the magnitudes AB, BC.
For, if CA, AB are not incommensurable, some magnitude will measure them.
Let it measure them, if possible, and let it be D. Since then D measures CA, AB,
therefore it will also measure the remainder BC,
But it measures AB also ; therefore D measures AB, BC
Therefore AB, BC are commensurable ; but they were also, by hypothesis, incommensurable : which is impossible.
Therefore no magnitude will measure CA, AB \ therefore CA, AB are incommensurable. [x. Def. i]
Similarly we can prove that AC, CB are also incom- mensurable.
Therefore AC\^ incommensurable with each of the magni- tudes AB, BC. ^
Next, let AChG incommensurable with one of the magni- tudes AB, BC ^
First, let it be incommensurable with AB ; I say that AB, BC are also incommensurable.
For, if they are commensurable, some magnitude will measure them.
Let it measure them, and let it be D.
Since then D measures AB, BC therefore it will also measure the whole AC
But it measures AB also ; therefore D measures CA, AB,
X. i6, 17] PROPOSITIONS 16, 17 41
Therefore CA, AB are commensurable ; but they were also, by hypothesis, incommensurable : which is impossible.
Therefore no magnitude will measure AB, BC ; therefore AB, BC are incommensurable. [x. Def. i]
Therefore etc.
Lemma.
If to any straight line there be applied a parallelogram deficient by a square figure , the applied parallelogram is equal to the rectangle contained by the segments of the straight line resulting from the application.
For let there be applied to the straight line AB the parallelogram AD deficient by the square figure DB\ I say that AD is equal to the rectangle contained by AC, CB.
This is indeed at once manifest ; for, since DB is a square, DC is equal to CB ;
and AD is the rectangle AC, CD, that is, the rectangle AC, CB.
Therefore etc.
If fl be the given straight line, and x the side of the square by which the apph'ed rectangle is to be deficient, the rectangle is equal to ax - x\ which is of course equal to x(a — x). The rectangle may be written xy, where x-¥y = a. Given the area x{a-x), or xy (whexie x+y = a), two different applications will give rectangles equal to this area, the sides of the defect being x or a-x {x or y) respectively ; but the second mode of expression shows that the rectangles do not differ in form but only in position.
Proposition 17,
If there be two unequal straight lines, and to the greater there be applied a parallelogram equal to the fourth part of the square on the less and deficient by a square figure, and if it divide it into parts which are commensurable in length, then 5 the square on the greater will be greater than the square on the less by the square on a straight line commensurable with the greater.
And, if the square on the greater be greater than the square an the less by the square on a straight line commensurable with
42 BOOK X [x. 17
10 the greater, and if there be applied to the greater a parallelogram equal to the fourth part of the square on the less and deficient by a square figure, it will divide it into parts which are com- mensurable in length.
Let Ay BC be two unequal straight lines, of which BC is 15 the greater, and let there be applied to BCz. parallel- ogram equal to the fourth part of the square on the less, A, that is, equal to the square on the half of y^, and deficient
20 by a square figure. Let this be the b f e rectangle BD, DC, [cf. Lemma]
and let BD be commensurable in length with DC\ I say that the square on BC is greater than the square on A by the square on a straight line commensurable with BC
25 For let BC be bisected at the point E, and let EF be made equal to DE,
Therefore the remainder DC is equal to BF, And, since the straight line BC has been cut into equal parts at E, and into unequal parts at /?,
30 therefore the rectangle contained by BD, DC, together with
the square on ED, is equal to the square on EC ; [n. 5]
And the same is true of their quadruples ;
therefore four times the rectangle BD, DC, together with
four times the square on DE, is equal to four times the square 35 on EC,
But the square on A is equal to four times the rectangle
BD, DC',
and the square on DF is equal to four times the square on
DE, for DF is double of DE. 40 And the square on BC is equal to four times the square
on EC, for again BC is double of CE.
Therefore the squares on A, DF are equal to the square
on BC,
so that the square on BC is greater than the square on A by 45 the square on DF,
It is to be proved that BC is also commensurable with DF, Since BD is commensurable in length with DC, therefore BC is also commensurable in length with CD. [x. 15]
X. 17] PROPOSITION 17 43
But CD is commensurable in length with CD, BF, for 50 CD is equal to BF. [x. 6]
Therefore BC is also commensurable in length with BF, CD, [X. 12]
so that BC is also commensurable in length with the remainder FD\ [X. 15]
55 therefore the square on BC is greater than the square on A by the square on a straight line commensurable with BC
Next, let the square on BC be greater than the square on A by the square on a straight line commensurable with BC, let a parallelogram be applied to BC equal to the fourth part 60 of the square on A and deficient by a square figure, and let it be the rectangle BD, DC
It is to be proved that BD is commensurable in length with DC
With the same construction, we can prove similarly that 65 the square on BC is greater than the square on A by the square on FD.
But the square on BC is greater than the square on A by the square on a straight line commensurable with BC
Therefore BC is commensurable in length with FD, 70 so that BC is also commensurable in length with the remainder, the sum of BF, DC [x. 15]
But the sum of BF, DC is commensurable with DC, [x. 6]
so that BC is also commensurable in length with CD ; [x. 12]
and therefore, separando, BD is commensurable in length
75 with DC [x- 15]
Therefore etc.
45. After saying literallv that **the square on BC is greater than the square on A by the square on /?/*," Euclid adds the equivalent expression with di/rarcu in its technical sense, ^ BF fifM r^f A fui^w hinnkrox rj AZ. As this is untranslatable in English except by a paraphrase in practically the same words as have preceded, I have not attempted to reproduce ik
This proposition gives the condition that the roots of the equation in x,
ax-x'^pf^ -, sayj,
are commensurable with a, or that x is expressible in terms of a and integral
numbers, i.e. is of the form — a. No better proof can be found for the fact
that Euclid and the Greeks used their solutions of quadratic equations for numerical problems. On no other assumption could an elaborate discussion of the conditions of incommensurability of the roots with given lengths or
44 BOOK X [x. 17
with a given number of units of length be explained In a purely geometrical solution the distinction between commensurable and incommensurable roots has no point, because each can equally easily be represented by straight lines. On the other hand, on the assumption that the nunurical sohiiion of quadratic equations was an important part of the system of the Greek geometers, the distinction between the cases where the roots are commensurable and incommensurable respectively with a given length or unit becomes of great importance. Since the Greeks had no means of expressing what we call an irrational number, the case of an equation with incommensurable roots could only be represented by them geometrically; and the geometrical representations had to serve instead of what we can express by formulae involving surds.
Euclid proves in this proposition and the next that, x being determined from the equation
x(a'-x) = - (i),
4
jp, (a ~ x) are commensurable in length when >/a* - ^, a are so, and incom- mensurable in length when Jc^ — ^, /i are incommensurable ; and conversely. Observe the similarity of his proof to our algebraical method of solving the equation, a being represented in the figure by BQ and x by CD^
EF^ED^--x
2
and jp (a ~ ^) + ( — x\ = — ^ by Eucl. 11. 5.
If we multiply throughout by 4,
/^(a-x)-\-^\^-xj =a«,
whence, by ( i ), ^ + (a — 2^)* = a\
or fl* — ^ = (a — 2x)\
and Vfl" — ^ = a - 2jir.
We have to prove in this proposition (i) that, if jc, {a-x) are commensurable in length, so are a, »J(^ — i^^ (2) that, if a, yld^ - //* are commensurable in length, so are x^ {a — x).
(i) To prove that a, a- 2x are commensurable in length Euclid employs several successive steps, thus.
[X. 15]
|
Since (a - |
^) |
r> X, |
a f^ X. |
||
|
But |
X ^ 2X. |
||||
|
Therefore |
a ^ 2x ^ (a- 2x), |
||||
|
That is. |
tf ^ Va'-^. |
||||
|
(2) Since |
a ' |
^ J^'- |
~^, |
a ^ a-2x, |
|
|
whence |
a ^ 2X. |
||||
|
But |
2X ^ x; |
||||
|
therefore |
a ^ X, |
||||
|
and hence |
{a- |
-x) *^ X. |
|
[X |
.6] |
|
[X. |
12] |
|
[X. |
IS] |
|
[X. |
15] |
|
[X |
.6] |
|
[X. |
,2] |
|
[X. |
•5] |
X. 17, i8] PROPOSITIONS 17, 18 45
It IS often more convenient to use the symmetrical form of equation in this and similar cases, viz.
The result with this mode of expression is that (i) if X ^y, then a r^ >Ja^-lf^; and (2) \{ a ^ \/i?-^, then x '^ y.
The truth of the proposition is even easier to see in this case, since
Proposition 18.
1/ there be two unequal straight lines, and to the greater there be applied a parallelogram equal to the fourth part of the square on the less and deficient by a square figure^ and if it divide it into parts which are incommensurable, the square on the greater will be greater than the square on t/ie less by the square on a straight line incommensurable with the greater,
Andy if the square on the greater be greater t/ian the square on the less by the square on a straight line incommensurable with the greater, and if there be applied to the greater a parallelogram eqtial to the fourth part of the square on the less and deficient by a square figure, it divides it into parts which are incommensurable.
Let A, BC be two unequal straight lines, of which BC is the greater,
and to BC let there be applied a parallelogram equal to the fourth part of the square on the less, A, and deficient by a square figure. Let this be the rect- angle BDy DC, [cf. Lemma before x. 17] g.. and let BD be incommensurable in length with DC\ I say that the square on BC is greater than the square on A by the square on a straight line incom- mensurable with BC,
For, with the same construction as before, we can prove similarly that the square on BC is greater than the square on A by the square on FD.
It is to be proved that BC is incommensurable in length with DF.
46 BOOK X [x. i8
Since BD is incommensurable in length with DC,
therefore BC is also incommensurable in length with CD.
[X. i6]
But DC is commensurable with the sum of BFy DC ; [x. 6] therefore BC is also incommensurable with the sum of BF, DC\ [X. 13]
so thatiffC is also incommensurable in length with the remainder FD. [x. 16]
And the square on BC is greater than the square on A by the square on FD ;
therefore the square on BC is greater than the square on A by the square on a straight line incommensurable with BC
Again, let the square on BC be greater than the square on A by the square on a straight line incommensurable with BC^ and let there be applied to BC a parallelogram equal to the fourth part of the square on A and deficient by a square figure. Let this be the rectangle BD, DC.
It is to be proved that BD is incommensurable in length with DC
For, with the same construction, we can prove similarly that the square on BC is greater than the square on A by the square on FD.
But the square on BC is greater than the square on A by the square on a straight line incommensurable with BC\ therefore BC is incommensurable in length with FD, so that BC is also commensurable with the remainder, the ^xxmoi BF, DC [x. 16]
But the sum of BF, DC is commensurable in length with DC\ [x. 6]
therefore BC is also incommensurable in length with DC,
• [x. 13] so that, separando, BD is also incommensurable in length with DC. [x. 16]
Therefore etc.
With the same notation as before, we have to prove in this proposition that
( 1 ) if (a - jc), X are incommensurable in length, so are a, Ja^ - ^, and
(2) if a, n/o* - ^ are incommensurable in length, so are (a - x), x. Or, with the equations
^]
I
X. 18, 19] PROPOSITIONS i8, 19 47
(i) if X y^yy then a ^ >/tf*-^, and (2) if a *^ >/a* - ^, then x ^ y.
The steps are exactly the same as shown under (i) and (2) of the last note, with ^ instead of ^^ except only in the lines "^ '^ 2x^' and **2;c '^ x^ which are unaltered, while, in the references, x. 13, 16 take the place of x. 12, 15 respectively.
[Lemma.
Since it has been proved that straight lines commen- surable in length are always commensurable in square also, while those commensurable in square are not always com- mensurable in length also, but can of course be either commensurable or incommensurable in length, it is manifest that, if any straight line be commensurable in length with a given rational straight line, it is called rational and commen- surable with the other not only in length but in square also, since straight lines commensurable in length are always commensurable in square also.
But, if any straight line be commensurable in square with a given rational straight line, then, if it is also commensurable in length with it, it is called in this case also rational and commensurable with it both in length and in square ; but, if again any straight line, being commensurable in square with a given rational straight line, be incommensurable in length with it, it is called in this case also rational but commensurable in square only.]
Proposition 19.
The rectangle contained by rational straight lines commen- surable in length is rational.
For let the rectangle AC he: contained by the rational straight lines AB, BC commensurable in length ; I say that -^C is rational.
For on AB let the square AD be de- scribed ; therefore AD is rational. [x. Def. 4]
And, since AB is commensurable in length with BC, while AB is equal to BD, therefore BD is commensurable in length with BC
48 BOOK X [x. 19
And, as BD is to BC, so is DA to AC, [vi. i]
Therefore DA is commensurable with AC. [x. u]
But DA is rational ; therefore AC \s also rational. [x. Def. 4]
Therefore etc.
There is a difficulty in the text of the enunciation of this proposition. The Greek runs to vwo pifnav fJuiJKei arvfJLfierptov icara riva rtav irpo€iprjfUvtav rpoTTcDv cv^cutfF ir€pi€ypiJL€vov opOoyioyiov prjrov Itrnv^ where the rectangle is said to be contained by *' rational straight lines commensurable in length in any of the aforesaid ways,^^ Now straight lines can only be commensurable in length in one way, the degrees of commensurability being commensurability in length and commensurability in square only. But a straight line may be rational in two ways in relation to a given rational straight line, since it may be either commensurable in lengthy or commensurable in square onfyy with the latter. Hence Billingsley takes Kara ro^a rwv irpo€iprffi€Vfav rpowtav with (nftiav, translating '* straight lines commensurable in length and rational in any of the aforesaid ways," and this agrees with the expression in the next proposition **a straight line once more rational in any of the aforesaid ways"; but the order of words in the Greek seems to be fatal to this way of translating the passage.
The best solution of the difficulty seems to be to reject the words "in any of the aforesaid ways " altogether. They have reference to the I-emma which immediately precedes and which is itself open to the gravest suspicion. It is very prolix, and cannot be called necessary ; it appears moreover in connexion with an addition clearly spurious and therefore relegated by Heiberg to the Appendix. The addition does not even pretend to be Euclid's, for it begins with the words ** for /le calls rational straight lines those...." Hence we should no doubt relegate the Lemma itself to the Appendix. August does so and leaves out the suspected words in the enunciation, as I have done.
Exactly the same arguments apply to the I^mma added (without the heading "Lemma") to x. 23 and the same words "in any of the aforesaid ways " used with " medial straight lines commensurable in length " in the enunciation of x. 24. The said Lemma must stand or fall with that now in question, since it refers to it in terms: "And in the same way as was explained in the case of rationals...."
Hence I have bracketed the Lemma added to x. 23 and left out the objectionable words in the enunciation of x. 24.
If p be one of the given rational straight lines (rational of course in the sense of x. Def. 3), the other can be denoted by kp, where k is, as usual, of the form mjn (where w, n are integers). Thus the rectangle is ifep', which is obviously rational since it is commensurable with p*. [x. Def. 4.]
A rational rectangle may have any of the forms a^, ha^, kA or A^ where fl, b are commensurable with the unit of length, and A with the unit of area.
Since Euclid is not able to use kp as a symbol for a straight line commensurable in length with p, he has to put his proof in a form corre- sponding to
p' \ kp^ = p: kp,
whence, p, kp being commensurable, p*, kp^ are so also. [x. 11]
X- 20, 2i] PROPOSITIONS 19—21 49
Proposition 20.
If a rational area be applied to a rational straight line, it produces as breadth a straight line rational and commensurable in length with the straight line to which it is applied.
For let the rational area AC h^ applied to AB, a straight line once more rational in any of the aforesaid ways, producing BC as breadth ; I say that BC is rational and commensurable in length with BA.
For on AB let the square AD be described ; therefore AD is rational. [x. Def. 4]
But ^C is also rational ; therefore DA is commensurable with AC.
And, as DA is to AC, so is DB to BC.
Therefore DB is also commensurable with BC ; [x. n] and DB is equal to BA ; therefore AB is also commensurable with BC.
But AB is rational ; therefore BC is also rational and commensurable in length with AB.
Therefore etc.
The converse of the last. If p is a rational straight line, any rational area
is of the form kp*. If this be "applied" to p, the breadth is kp commensurable
in length with p and therefore rational. We should reach the same result if
we applied the area to another rational straight line o-. The breadth is then
k^ kp* m J ,,
-^= -^.a = — k.crorkfr. say.
Proposition 21.
The rectangle contained by rational straight lines commen- surable in square only is irrational^ and the side of the square equal to it is irrational. Let the latter be called medial.
For let the rectangle ^C be contained by the rational straight lines ABy BC commensurable in square only ;
H. E. in. 4
so
BOOK X
[X.
I say that AC is irrational, and the side of the square equal
to it is irrational ;
and let the latter be called medial.
For on AB let the square AD he described ; therefore AD is rational. [x. Def. 4]
And, since AB is incommensurable in length with BC,
for by hypothesis they are commensurable in square only,
while AB is equal to BD, therefore DB is also incommensurable in length with BC.
And, as DB is to BC, so is AD to AC; [vi. 1]
therefore DA is incommensurable with AC. [x. "]
But DA is rational ; therefore -^C is irrational,
so that the side of the square equal to ^C is also irrational.
[x. Def. 4] And let the latter be called medial.
Q. E. D.
A media/ straight line, now defined for the first time, is so called because it is a mean proportional between two rational straight lines commensurable in square only. Such straight lines can be denoted by p, p Jk. A medial
straight line is therefore of the form Jp^^k or k*p. Euclid's proof that this is irrational is equivalent to the following. Take p, pjk commensurable in square only, so that they are incommensurable in length.
Now P''P>J^ = P*'P^JK
whence [x. 11] p^Jk is incommensurable with p* and therefore irrational [x. Def. 4], so that Jp^Jk is also irrational \tbid\
A medial straight line may evidently take either of the forms Ja^JB or \lABy where of course B is not of the form l^A.
Lemma.
If there be two straight lines, then, as the first is to the second, so is the square on the first to the rectangle contained by the two straight lines.
Let FE, EG be two straight lines. °
I say that, as FE is to EG, so is the square on FE to the rectangle FEy EG.
X. 2i, 22]
PROPOSITIONS 21, 22
51
For on FE let the square DF be described, and let GD be completed.
Since then, as FE is to EG^ so is FD to DG^ [vi. i]
and FD is the square on FE,
and Z>G^ the rectangle DE, EG, that is, the rectangle FE^ EG, therefore, as FE is to EG, so is the square on FE to the rectangle FE, EG.
Similarly also, as the rectangle GE, EF is to the square on EF, that is, as GD is to FD, so is GE to EF.
Q. E. D.
If fl, b be two straight lines,
a'.h-c^\ ab.
Proposition 22.
7"^ sqtuire on a medial straight line, if applied to a rational straight line, produces as breadth a straight line rational and incommensurable in length with that to which it is applied.
Let A be medial and CB rational,
and let a rectangular area BD equal to the square on A be applied to BC, producing CD as breadth ;
I say that CD is rational and incom- mensurable in length with CB.
For, since A is medial, the square on it is equal to a rectangular area contained by rational straight lines commensurable in square only.
[X. 21]
Let the square on it be equal to GF.
But the square on it is also equal to BD ; therefore BD is equal to GF.
But it is also equiangular with it ; and in equal and equiangular parallelograms the sides about the equal angles are reciprocally proportional ; [vi. 14]
therefore, proportionally, as BC is to EG, so is EF to CD.
Therefore also, as the square on BC is to the square on EG, so is the square on EF to the square on CD. [vi. 22]
4—2
|
B A |
Q ) 1 |
||
|
( |
) |
I F |
52 BOOK X [x. 2 2
But the square on CB is commensurable with the square on EG, for each of these straight lines is rational ;
therefore the square on EF is also commensurable with the square on CD. [x. n]
But the square on EF is rational ; therefore the square on CD is also rational ; [x. Def. 4]
therefore CD is rational.
And, since EFxs incommensurable in length with EG,
for they are commensurable in square only,
and, as EF is to EG, so is the square on EF to the rectangle FE, EG, [Lemma]
therefore the square on EF is incommensurable with the rectangle FE, EG. [x. n]
But the square on CD is commensurable with the square on EFy for the straight lines are rational in square ; and the rectangle DC, CB is commensurable with the rect- angle FE, EG, for they are equal to the square on A ; therefore the square on CD is also incommensurable with the rectangle DC, CB. [x. 13]
But, as the square on CD is to the rectangle DC, CB, so is DC to CB ; [Lemma]
therefore DC is incommensurable in length with CB. [x. 11]
Therefore CD is rational and incommensurable in length with CB.
Q. E. D.
Our algebraical notation makes the result of this proposition almost self- evident. We have seen that the square of a medial straight line is of the form Jk,p\ If we "apply'' this area to another rational straight line <r, the
breadth is '^-?\
(T
It 2
This is equal to \ .<T=,Jk. — <T, where ///, n are integers. The latter
straight line, which we may express, if we please, in the form Jk\ or, is clearly commensurable with <r in square only, and therefore rational but incom- mensurable in length with <r.
Euclid's proof, necessarily longer, is in two parts.
Suppose that the rectangle ^Jk . p^ = a'.x.
Then (i) <r:p=^k.p:x, [vi. 14]
whence <t^ : p^ = J^p^ : x^. [vi. 22]
But cr' <^ p% and therefore ^p' ^ a::*. [^- ' ']
X, 22, 23]
PROPOSITIONS 22, 23
53
And k^ is rational ; therefore j:*, and therefore jc, is rational.
(2) Since ^^ . /> '^ p, Jk.psj p.
But [Lemma] Jk.p:p = kp^: Jk,p\
whence V ^ J^ • p'-
But Jk.p^~ era;, and ^p' '^ :r' (from above) ; therefore c^^^<tx \
and, since :)^ i ax =^ x : cr^
X s^ a.
[x. Def. 4]
[X. II]
[X. 13] [Lemma]
D F
Proposition 23.
A straight line commensurable with a medial straight line is medial.
Let A be medial, and let B be commensurable with A ; I say that B is also medial.
For let a rational straight line CD
be set out, — ■
and to CD let the rectangular area CE equal to the square on A be applied, producing ED as breadth ; therefore ED is rational and incommen- surable in length with CD. [x. 22]
And let the rectangular area CF equal to the square on B be applied to CD, producing DF as breadth.
Since then A is commensurable with B, the square on A is also commensurable with the square on B,
But EC is equal to the square on A, and CF is equal to the square on B ; therefore EC is commensurable with CF,
And, as EC is to CF, so is ED to DF\ therefore ED is commensurable in length with DF.
But ED is rational and incommensurable in length with DC\
therefore DF is also rational [x. Def 3] and incommensurable in length with DC. [x. 13]
Therefore CD, DF are rational and commensurable in square only.
[VI. I] [X. II]
54 BOOK X [x. 23
But the straight line the square on which is equal to the rectangle contained by rational straight lines commensurable in square only is medial ; ♦ [x. 21]
therefore the side of the square equal to the rectangle CD^ DF is medial.
And B is the side of the square equal to the rectangle CD,DF\ therefore B is medial.
PoRiSM. From this it is manifest that an area commen- surable with a medial area is medial.
[And in the same way as was explained in the case of rationals [Lemma following x. 18] it follows, as regards medials, that a straight line commensurable in length with a medial straight line is called medial and commensurable with it not only in length but in square also, since, in general, straight lines commensurable in length are always commensurable in square also.
But, if any straight line be commensurable in square with a medial straight line, then, if it is also commensurable in length with it, the straight lines are called, in this case too, medial and commensurable in length and in square, but, if in square only, they are called medial straight lines commen- surable in square only.]
As explained in the bracketed passage following this proposition, a straight line commensurable with a medial straight line in square otily^ as well as a straight line commensurable with it in length, is medial.
Algebraical notation shows this easily.
If Irp be the given straight line, \lrp is a straight line cortimensurable
in length with it and ^\ . Irp a straight line commensurable with it in square only.
But Ap and V^P ^'"^ ho\)^ rational [x. Def. 3] and therefore can be expressed by p', and we thus arrive at trp^ which is clearly medial.
Euclid's proof amounts to the following.
Apply both the areas Jk,p^ and k^Jk.p^ (or XJk.p^) to a rational straight line <r.
The breadths Ji . - and A V't • - (or A^^ . ^ j are in the ratio of the
areas Jk,p^ and X^Jk.p^ (or kjk.p^) themselves and are therefore com- mensurable.
Now [x. 22] J^ . — is rational but incommensurable with a.
Therefore A*^^ . — f or A^^ . - j is so also;
X. 23, 24] PROPOSITIONS 23, 24 55
whence the area >?Jk . p* (or X^^ . p*) is contained by two rational straight
lines commensurable in square only, so that X^V (or V^ • '^ p) 's a medial straight line.
It is in the Porism that we have the first mention of a medial area. It is the area which is equal to the square on a medial straight line^ an area, there- fore, of the form k^p\ which is, as a matter of fact, arrived at, though not named, before the medial straight line itself (x. 21).
The Porism states that Xk^p^ is a medial area, which is indeed obvious.
Proposition 24.
The rectangle contained by medial straight lines commen- surable in length is medial.
For let the rectangle AC he contained by the medial straight lines AB, BC which are commensurable in length ;
I say that AC is medial.
For on AB let the square AD be described ;
therefore AD is medial
And, since AB is commensurable in length with BC,
while AB is equal to BD,
therefore DB is also commensurable in length with BC;
so that DA is also commensurable with AC. [vi. i, x. n]
But DA is medial ;
therefore AC is also medial. [x. 23, Por.]
Q. E. D.
There is the same difficulty in the text of this enunciation as in that of X. 19. The Greek says "medial straight lines commensurable in length in any of the aforesaid ways " ; but straight lines can only be commensurable in length in one way, though they can be medial in two ways, as explained in the addition to the preceding proposition, i.e. they can be either commensurable in length or commensurable in square only with a given medial straight line. For the same reason as that explained in the note on x. 19 I have omitted " in any of the aforesaid ways " in the enunciation and bracketed the addition to X. 23 to which it refers.
Irp and yJrp are medial straight lines commensurable in length. The
rectangle contained by them is X^V» which may be written l^p^ and is there- fore clearly medial.
Euclid's proof proceeds thus. Let x^ \x be the two medial straight lines commensurable in length.
Therefore oi?\x .\x-X'.\x.
$6
BOOK X
[x. 24. *S [X. ,.]
But ^ ^ Xjc, so that x^^x.Xx.
Now jc" is medial [x. 21] ; therefore jc . A;r is also medial. [x. 23, Por.]
We may of course write two medial straight lines commensurable in length in the forms mJ^p, n^p; and these may either be mJaJB^ nJaJB, or mijAB, ni/AB.
Proposition 25*
TAe rectangle contained by medial straight lines comment surable in square only is either rational or medial.
For let the rectangle AC he contained by the medial straight lines AB, BC which are commensurable in square only ; I say that ACis either rational or medial.
For on AB, BC let the squares AD, BE be described ; therefore each of the squares ADy BE is medial.
Let a rational straight line EG be set out,
to EG let there be applied the rectangular parallelogram G/f equal to AD, producing E// as breadth, to //M let there be applied the rectangular parallelogram MK equal to AC, producing //K sls breadth, and further to AW let there be similarly applied NL equal to BE, producing KL as breadth ; therefore E//, HK, KL are in a straight line.
Since then each of the squares AD, BE is medial, and AD is equal to GH, and BE to NL, therefore each of the rectangles GH, NL is also medial.
And they are applied to the rational straight line EG ; therefore each of the straight lines EH, KL is rational and
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L |
incommensurable in length with EG,
And, since -^Z? is commensurable with BE^
therefore GH is also commensurable with NL, And, as GH is to NL, so is EH to KL ;
therefore EH is commensurable in length with KL.
[x. 22]
[VI. ,]
[X. ,rj
X. 25l PROPOSITIONS 24, 25 57
Therefore FH, KL are rational straight lines commen- surable in length ; therefore the rectangle FH, KL is rational. [x. 19]
And, since DB is equal to BA, and OB to BC, therefore, as DB is to BC, so is AB to BO.
But, as DB is to BC, so is DA to AC, [vi. i]
and, as AB is to BO, so is y^C to CO ; \id:\
therefore, as DA is to ^ C, so is -^ C to CO.
But AD is equal to GH, AC to i^/A" and CO to 7VZ ;
therefore, as GH is to MK, so is J/A' to NL ;
therefore also, as FH is to /^A', so is HK to A'Z ; [vi. i, v. n]
therefore the rectangle FH, KL is equal to the square on HK.
[vi. 17] But the rectangle FH, KL is rational ;
therefore the square on HK is also rational.
Therefore HK is rational.
And, if it is commensurable in length with FG,
HN is rational ; [x. 19]
but, if it is incommensurable in length with FG,
KH, HM 2lx^ rational straight lines commensurable in square only, and therefore HN is medial. [x. 21]
Therefore HN is either rational or medial.
But HN is equal to AC\
therefore -^C is either rational or medial.
Therefore etc.
Two medial straight lines commensurable in square only are of the form
The rectangle contained by them is ^\ . >^V- Now this is in general medial; but, if ^/A. = k Jk, the rectangle is kk'p^, which is rational,
Euclid's argument is as follows. Let us, for convenience, put x for ^*p, so that the medial straight lines are x, ^k.x.
Form the areas ^, x . JX.x, Ajc*, and let these be respectively equal to o-w, fw, aw, where o- is a rational straight line.
Since x^, Xx* are medial areas, so are «n/, <ra/, whence u, w are respectively rational and '^ o-.
58 BOOK X [x. 25, 26
But x^'^Xx^, so that <ru ^ aw, or u f^ w (i).
Therefore, «, w being both rational, uw is rational (2).
Now x':Jk.x'=^\.a^:\x'
or a-u :<rv = (rv : aWy
so that u:Vj=v:w,
and uw = fr^.
Hence, by (2), »*, and therefore Vy is rational (3).
Now (a) if zf r\ a-y arv or J\ . x^ is rational]
{P) if t^ w cr, so that v ^^ a-y <w or Jk . x^ is media/.
Proposition 26. A medial area does not exceed a medial area by a rational
area.
For, if possible, let the medial area AB exceed the medial area AC by the rational area DB,
and let a rational straight line jEJ^ be set out ;
to BjP^ let there be applied the rectangular parallelogram J^// equal to AB, producing £// as breadth,
and let the rectangle FG equal to ^C be subtracted ; therefore the remainder BD is equal to the remainder K//.
But DB is rational ; therefore KH is also rational.
Since, then, each of the rectangles AB, AC is medial, and AB is equal to F//, and ^C to FGy therefore each of the rectangles F//, FG is also medial.
And they are applied to the rational straight line EFy therefore each of the straight lines HEy EG is rational and incommensurable in length with EF. [x. 22]
And, since \JDB is rational and is equal to KH, therefore] KH is [also] rational ; and it is applied to the rational straight line EF\
X. 26] PROPOSITIONS 25, 26 59
therefore GH is rational and commensurable in length with EF. [x. 20]
But EG is also rational, and is incommensurable in length with EF\
therefore EG is incommensurable in length with GH. [x. 13]
• And, as EG is to GH, so is the square on EG to the rectangle EG, GH ;
therefore the square on EG is incommensurable with the rectangle EG, GH, [x. n]
But the squares on EG, GH are commensurable with the square on EG, for both are rational ; ^
and twice the rectangle EG, GH is commensurable with the rectangle EG, GH, for it is double of it ; [x. 6]
therefore the squares on EG, GH are incommensurable with twice the rectangle EG, GH \ [x. 13]
therefore also the sum of the squares on EG, GH and twice the rectangle EG, GH, that is, the square on EH [n. 4], is incommensurable with the squares on EG, GH. [x. 16]
But the squares on EG, GH are rational ; therefore the square on EH is irrational. [x. Def. 4]
Therefore EH is irrational.
But it is also rational : which is impossible.
Therefore etc.
Q. E. D.
" Apply " the two given medial areas to one and the same rational straight
line p. They can then be written in the form /» . k^p, p . \^p.
The difference is then (J^- J^)p^; and the proposition asserts that this cannot be rational, i.e. (Jk — JX) cannot be equal to k'. Cf. the proposition corresponding to this in algebraical text-books.
To make Euclid's proof clear we will put x for k^p and y for \^p.
Suppose p {x -y) = pz,
and, if possible, let pz be rational, so that z must be rational and ^ p ..(i).
Since px, py are medial,
X and y are respectively rational and s^ p (2).
From (i) and (2), y s^ z.
Now y : z =y :yz,
so that y ^^ yz.
6o BOOK X [x. 26, 27
But y-f-2'^y,
and 2yz ^ yz.
Therefore j^ + «" s^ 2yzy
whence (y + zf s^ (y + «'),
or a;* w (y + 0*).
And (y + s^) is rational ; therefore jc", and consequently ^, is irrational.
But, by (2), :x: is rational : which is impossible.
Therefore pz is not rational.
Proposition 27.
To find medial straight lines commensurable in square only which contain a rational rectangle.
Let two rational straight lines Ay B commensurable in square only be set out ;
let C be taken a mean proportional between Ay By [VI. 13]
and let it be contrived that,
as-y^ is to By so is C to D. [vi. 12]
Then, since Ay B are rational and com- mensurable in square only, the rectangle Ay By that is, the square on C [vi. 17], is medial. [x. 21]
Therefore C is medial. [x. 21]
And since, as A is to By so is C to /?, and Ay B are commensurable in square only, therefore C, D are also commensurable in square only. [x. n]
And C is medial ; therefore D is also medial. [x. 23, addition]
Therefore C, D are medial and commensurable in square only.
I say that they also contain a rational rectangle.
For since, as A is to By so is C to /?, therefore, alternately, as A is to C so is B to D, [v. 16]
But, as A is to C, so is C to ^ ; therefore also, as C is to By so is j5 to Z? ; therefore the rectangle C, D is equal to the square on B.
X. 27, 28] PROPOSITIONS 26—28 61
But the square on B is rational ; therefore the rectangle C, D is also rational.
Therefore medial straight lines commensurable in square only have been found which contain a rational rectangle.
Q. E. D. Euclid takes two rational straight lines commensurable in square only, say
p. k\.
Find the mean proportional, i.e. f^p.
Take x such that p:k^p = frp:x (i).
This gives x = Irp, and the lines required are ^p, ^*p. For (a) ^p is medial. And {P)y by (i), since p ^^ k^p,
k*p ^^- ^*p,
whence [addition to x. 23], since Irp is medial,
k^p is also medial. The medial straight lines thus found may take either of the forms
(0 ^7JB, y^ or (2) VaB, JT^^.
Proposition 28.
To find medial straight lines commensurable in square only which contain a medial rectangle.
Let the rational straight lines A, B, C commensurable in square only be set out ;
let D be taken a mean proportional between Ay By [vi. 13] and let it be contrived that,
as B is to C, so is D to E, [vi. 12]
A
B °-
C-
E-
Since Ay B are rational straight lines commensurable in square only,
therefore the rectangle Ay By that is, the square on D [vi. 17], IS medial. [x. 21]
62 BOOK X [x. 28
Therefore D is medial. [x. 21]
And since By C are commensurable in square only,
and, as B is to C, so is D to E^
therefore D, E are also commensurable in square only. [x. n] But D is medial ;
therefore E is also medial. [x. 23, addition]
Therefore D, E are medial straight lines commensurable in square only.
I say next that they also contain a medial rectangle.
For since, as B is to C, so is D to Ey therefore, alternately, as B is to /?, so is C to E. [v. 16]
But, as i9 is to /?, so is /? to ^ ;
therefore also, as D is to A^ so is C to £* ;
therefore the rectangle ^, C is equal to the rectangle D, E.
[vi. 16] But the rectangle A, C is medial ; [x. 21]
therefore the rectangle /?, E is also medial.
Therefore medial straight lines commensurable in square only have been found which contain a medial rectangle.
Q. E. 1).
Euclid takes three straight lines commensurable in square only, i.e. of the
form p, ^V' ^ P> ^"cl proceeds as follows.
Take the mean proportional to p, ^^p, i.e. Ir^ Then take x such that
/tV:^*P = ^*p:^ (i),
so that X = k^pl^,
frpy X^pjfr are the required medial straight lines.
For /rp is medial.
Now, by (i), since k^p *^ A^p,
^p '^ X, *
whence x is also medial [x. 23, addition], while ^^ k*p. Next, by (i), \^p:x = k^p : k^p
= ^P •• Pi whence x,k^p = xV» which is medial.
The straight lines ^V> ^^pjk of course take different forms according as the original straight lines are of the forms (i) a, ^B^ JQ (2) ^M, JB, JC, (3) ^A, b, VC, and (4) si A, JB, c.
X. 28, Lemma i] PROPOSITION 28 63
E.g. in case (i) they are Ja^B, \J ^j^^
in case (2) they are and so on.
^Jab^ J^^
Lemma i. To find two sqimre numbers such that their sum is also square.
Let two numbers AB, BC be set out, and let them be either both even or both odd.
Then since, whether an even a d 6 b
number is subtracted from an
even number, or an odd number from an odd number, the remainder is even, [ix. 24, 26]
therefore the remainder AC is even.
Let -^C be bisected at /?.
Let AB, BC also be either similar plane numbers, or square numbers, which are themselves also similar plane numbers.
Now the product of AB, BC together with the square on CD is equal to the square on BD. [11. 6]
And the product of AB, BC is square, inasmuch as it was proved that, if two similar plane numbers by multiplying one another make some number, the product is square, [ix. i]
Therefore two square numbers, the product of AB, BC, and the square on CD, have been found which, when added together, make the square on BD.
And it is manifest that two square numbers, the square on BD and the square on CD, have again been found such that their difference, the product of AB, BC, is a square, whenever AB, BC are similar plane numbers.
But when they are not similar plane numbers, two square' numbers, the square on BD and the square on DC, have been found such that their difference, the product of AB, BC, is not square.
Q. E. D.
Euclid's method of forming right-angled triangles in integral numbers, already alluded to in the note on i. 47, is as follows.
Take two similar plane numbers, e.g. mn/i^, mnq^, which are eit/ier both even or both odd, so that their difference is divisible by 2.
64 BOOK X [Lemmas i, 2
Now the product of the two numbers, or tf^f^ff^ is square, [ix. i]
and, by 11. 6,
mnf . mruj^ "•" ( ~ ) ^ ( — ) »
so that the numbers mnp^, J {mn/^ - mnq^) satisfy the condition that the sum of their squares is also a square number.
It is also clear that J (/««/* + /««/), mnpq are numbers such that the difference of their squares is also square.
Lemma 2.
To find two square numbers such t/icU their sum is not square.
For let the product of ABy BC, as we said, be square, and CA even, and let CA be bisected by D,
E A 1 ' 1 ± ^
AG H D
It is then manifest that the square product of AB, BC together with the square on CD is equal to the square on BD,
[See Lemma i] Let the unit DE be subtracted ;
therefore the product of AB, BC together with the square on CE is less than the square on BD,
I say then that the square product of AB, BC together with the square on CE will not be square.
For, if it is square, it is either equal to the square on BE^ or less than the square on BE, but cannot any more be greater, lest the unit be divided.
First, if possible, let the product of AB, BC together with the square on CE be equal to the square on BE, and let GA be double of the unit DE.
Since then the whole ^C is double of the whole CD, and in them AG is double of DE,
therefore the remainder GC is also double of the remainder EC; therefore GC is bisected by E.
Therefore the product of GB, BC together with the square on CE is equal to the square on BE. [n. 6]
But the product of AB, BC together with the square on CE is also, by hypothesis, equal to the square on BE ;
Lemma 2] LEMMAS TO PROPOSITIONS 29, 30 65
therefore the product of GB^ BC together with the square on CE is equal to the product of AB, BC together with the square on CE.
And, if the common square on CE be subtracted, it follows that AB is equal to GB : which is absurd.
Therefore the product of AB, BC together with the square on CE is not equal to the square on BE.
I say next that neither is it less than the square on BE.
For, if possible, let it be equal to the square on BE, and let If A be double of DE.
Now it will again follow that I/C is double of CE; so that Clf has also been bisected at E, and for this reason the product of IfB, BC together with the square on EC is equal to the square on BE. [11. 6]
But, by hypothesis, the product of AB, BC together with the square on CE is also equal to the square on BE.
Thus the product of /fB, BC together with the square on CE will also be equal to the product of AB, BC together with the square on CE :
which is absurd.
Therefore the product of AB, BC together with the square on CE is not less than the square on BE.
And it was proved that neither is it equal to the square on^^.
Therefore the product of AB, BC together with the square on CE is not square.
Q. E. D.
We can, of course, write the identity in the note on Lemma i above (p. 64) in the simpler form
where, as before, mf^, m^ are both odd or both even. Now, says Euclid,
m^ . m^ + { — — - I J is not a square number.
This is proved by reductio ad absurdum.
H. £. III. 5
66 BOOK X [Lemma 2, x. 29
The number is clearly less than m^ . mf + \-S-I- — T\ ^ i,e. less than
If then the number is square, its side must be greater than, equal to, or less than i— — — ^- i j, the number next less than — — — ^.
But (i) the side cannot be > i— ^— i j without being equal to
-^ —^ since they are consecutive numbers.
If then mf.mf^ {»il^ _ .^ is also equal to (^^±^ _ ,)'.
we must have (w/* - 2) «^' = «/" . w^,
or w^" - 2 = mjf^ :
which is impossible.
(3) If .;,.,^.(!!5^^,y<(^i±^_,y.
suppose It equal to ( -^ -^ - r\ .
But [... 6] {mf-2r)m4'^(^^-r)"^(^^^-r)\ Therefore
which is impossible.
Hence all three hypotheses are false, and the sum of the squares mf, mf and ( F-^T _ j\ jg ^^ square.
Proposition 29.
To find two rational straight lines commensurable in square only and such that the square on the greater is greater than the square on the less by the square on a straight line commen- surable in length with the greater.
For let there be set out any rational straight line AB, and two square numbers CD, DE such that their difference CE is not square ; [Lemma i]
let there be described on AB the semicircle AFB,
\
X, 29] PROPOSITION 29 67
and let it be contrived that,
as DC is to CE^ so is the square on BA to the square on AF. [x. 6, Por.]
Let FB be joined.
Since, as the square on BA is to the square on AFy so is DC to CE^ therefore the square on BA has to the square on AF the ratio which the
number DC has to the number CE ; g g P
therefore the square on BA is com- mensurable with the square on AF. [x. 6]
But the square on AB is rational ; [x. Def. 4]
therefore the square on AF is also rational ; [i^/.]
therefore AF is also rational.
And, since DC has not to CE the ratio which a square number has to a square number,
neither has the square on BA to the square on AF the ratio which a square number has to a square number ; therefore ABv& incommensurable in length with AF. [x. 9]
Therefore BA, AF are rational straight lines commen- surable in square only.
And since, as DC is to CE, so is the square on BA to the square on AF,
therefore, convertendo, as CD is to DE, so is the square on AB to the square on BF. [v. 19, Por., in. 31, i. 47]
But CD has to DE the ratio which a square number has to a square number ;
therefore also the square on AB has to the square on BF the ratio which a square number has to a square number ; therefore AB is commensurable in length with BF. [x. 9]
And the square onABxs equal to the squares on AF, FB\
therefore the square on AB is greater than the square on AF by the square on BF commensurable with AB.
Therefore there have been found two rational straight lines BA, -^4/^ commensurable in square only and such that the square on the greater AB is greater than the square on the less AF by the square on BF commensurable in length with AB.
Q. E. D.
5—2
68 BOOK X [x. 29, 30
Take a rational straight line p and two numbers m\ «" such that (if^ - «') is not a square.
Take a straight line x such that
m^\m*-n^ = f^\o^ (i),
whence ^ = 5— p',
and x=p Vi ->^, where >^ = - .
Then p, pVi ->P are the straight lines required.
It follows from (i) that ^ ^ p^
and X is rational, but x ^^ p.
By (i), canvertendo, m^:n* = p* :p^- x^y
so that Jp^ — oc^ ^ p, and in fact = kp.
According as p is of the form a or J A, the straight lines are (i) a, Jc^-l^ or (2) V^, slA-I^A.
Proposition 30.
To find two rational straight lines commensurable in square only and such that the square on the greater is greater than the square on the less by the square on a straight line incom-- mensurable in length with the greater.
Let there be set out a rational straight line AB, and two square numbers CE^ ED such that their sum CD is not square ; [Lemma 2]
let there be described on AB the semicircle AFB, let it be contrived that, as DC is to CE, so is the square on BA to the square on AFy
[x. 6, Por.] and let FB be joined.
Then, in a similar manner to the preceding, we can prove that BA, AF are rational straight lines commensurable in square only.
And since, as DC is to CE, so is the square on BA to the square on AF^
therefore, convertendo, as CD is to DE^ so is the square on AB to the square on BF. [v. 19, Por., in. 31, i. 47]
But CD has not to DE the ratio which a square number has to a square number ;
X. 3o» 31] PROPOSITIONS 29—31 69
therefore neither has the square on AB to the square on BF the ratio which a square number has to a square number ; therefore AB is incommensurable in length with BF. [x. 9]
And the square on AB is greater than the square on AF by the square on FB incommensurable with AB.
Therefore AB, AF are rational straight lines commen- surable in square only, and the square on AB is greater than the square on AF by the square on FB incommensurable in length with AB.
Q. E. D. In this case we take m\ «" such that w' + «" is not square. Find X such that m^ + n^ :m^ = p^ : ar*,
whence ^ = —^ = p' ,
or X == , ■ , where ^ = — .
Then p, . ^ satisfy the condition.
VI +^ The proof is after the manner of the proof of the preceding proposition and need not be repeated.
According as p is of the form a or J A, the straight lines take the
fl" p, that is, tf, Vtf' — By or (2) ^A, i^A - B and
Proposition 31.
To find two medial straight lines commensurable in square only^ containing a rational rectangle, and such that the square on the greater is greater than the square on the less by the square on a straight line commensurable in length with the grecUer.
Let there be set out two rational straight lines A, B commensurable in square only and such that the square on A, being the greater, is greater than the square on B the less by the square on a straight line commensurable in length with A.
[x. 29]
And let the square on C be equal to the rectangle A, B. a b c d
Now the rectangle A, B is medial ; [x, 21] therefore the square on C is also medial ; therefore C is also medial. [x. 21]
70
BOOK X [x. 31
Let the rectangle C, D be equal to the square on B.
Now the square on B is rational ; therefore the rectangle C, D is also rational.
And since, as A is to B, so is the rectangle A, B to the square on B^
while the square on C is equal to the rectangle A, By and the rectangle C, D is equal to the square on B, therefore, as A is to B, so is the square on C to the rectangle CD.
But, as the square on C is to the rectangle C, /?, so is C toD; therefore also, as A is to B, so is C to D.
But A is commensurable with B in square only ; therefore C is also commensurable with Z? in square only. [x. 1 1]
And C is medial ; therefore D is also medial. [x. 23, addition]
And since, as u4 is to B, so is C to /?, and the square on A is greater than the square on B by the square on a straight line commensurable with A, therefore also the square on C is greater than the square on Z? by the square on a straight line commensurable with C
[X. 14]
Therefore two medial straight lines C, D, commensurable in square only and containing a rational rectangle, have been found, and the square on C is greater than the square on D by the square on a straight line commensurable in length with C.
Similarfy also it can be proved that the square on C exceeds the square on Z? by the square on a straight line incommensurable with C, when the square on A is greater than the square on B by the square on a straight line incom- mensurable with A. [x. 30]
I. Take the rational straight lines commensurable in square only found in X. 29, i.e. p, p Vi - ^.
Take the mean proportional p (i - ^)* and x such that p(i->^)^:pVir^ = pVr^:^.
Then p (i - ^)*, :r, or p (i - ^)*, p (i - ^)' are straight lines satisfying the given conditions.
X. 31, 32] PROPOSITIONS 31, 32 71
For (a) p»Vi -^ is a medial area, and therefore p(i -^)* is a medial
straight line (i);
and jc . p (i - I^y = /»' (i - ^) and is therefore a rational area.
(/9) p, p(i - >^)*, pVi -^, Jp are straight lines in continued proportion, by construction.
Therefore p :p Vi ->^ = p(i -^fe*)* : Jc (2).
(This Euclid has to prove in a somewhat roundabout way by means of the lemma after x. 21 to the effect that a : b = ab : i^,)
From (2) it follows [x. 1 1] that xr^p{i- ^)* ; whence, since p (i - ^)* is
medial, jf or p (i — ^)* is medial also.
(y) From (2), since p, p Vi ->^ satisfy the remaining condition of the
problem, p(i->^)*, p(i->fe*)* do so also [x. 14].
According as p is of the form a or JA^ the straight lines take the forms
(i) Ja^<^-lf',
or (2) ijA{A-f^A), ^"'^^
t/AiA-k^A)'
II. To find medial straight lines commensurable in square only contain- ing a rational rectangle, and such that the square on one exceeds the square on the other by the square on a straight line incomfmnsurabie with the former, we simply begin with the rational straight lines having the corresponding
property [x. 30], viz. p, . , and we arrive at the straight lines
p p
According as p is of the form a or JA^ these (if we use the same transformation as at the end of the note on x. 30) may take any of the forms
(1) -Jlj^^, '^-^
or (2) XJA{A-B),
or V^(^-^),
A- B *>JA{A-By
VA{A-t»y
Proposition 32.
To find two medial straight lines commensurable in square only, containing a medial rectangle, and such that the square on the greater is greater than the square on the less by the square on a straight line commensurable with the greater.
72 BOOK X [x. 32
Let there be set out three rational straight lines A, B, C commensurable in square only, and such that the square on A is greater than the square on C by the square on a straight line commensurable with A, • [x. 29]
and let the square on D be equal to the rectangle A, B.
A
D
B
E
0
Therefore the square on D is medial ; therefore D is also medial. [x. 21]
Let the rectangle /?, E be equal to the rectangle B, C.
Then since, as the rectangle A,B isto the rectangle B, C, so is A to C,
while the square on D is equal to the rectangle A, B, and the rectangle /?, E is equal to the rectangle By C, therefore, as A is to C, so is the square on D to the rectangle D,E.
But, as the square on D is to the rectangle /?, E, so is D toE; therefore also, as -^4 is to C, so is D to E.
But A is commensurable with C in square only ; therefore Z? is also commensurable with E in square only. [x. 1 1]
But D is medial ; therefore E is also medial. [x. 23, addition]
And, since, as A is to C, so is Z? to E, while the square on A is greater than the square on C by the square on a straight line commensurable with y4, therefore also the square on D will be greater than the square on E by the square on a straight line commensurable with D.
[x. 14]
I say next that the rectangle Z?, E is also medial.
For, since the rectangle B, C is equal to the rectangle /?, E, while the rectangle B, C is medial, [x- 21]
therefore the rectangle /?, E is also medial.
Therefore two medial straight lines /?, E, commensurable in square only, and containing a medial rectangle, have been found such that the square on the greater is greater than the
X. 32] PROPOSITION 32 73
square on the less by the square on a straight line commen- surable with the greater.
Similarly again it can be proved that the square on D is greater than the square on E by the square on a straight line incommensurable ^vith Z?, when the square on A is greater than the square on C by the square on a straight line incommensurable with A. [x. 30]
I. Euclid takes three straight lines of the form p, p »J\ pVi-^**,
takes the mean proportional pX* between the first two (i),
and then finds x such that
pX*:pX* = p\/r^^:^ (2),
whence x = pX* vi - ^,
and the straight lines pX*, pX* >/i — ^ satisfy the given conditions. Now (a) pX* is medial. (fi) We have, from (i) and (2),
p:pji-^ = p\^:x (3),
whence x r^ pX* ; and x is therefore medial and ^^ pX*.
(y) x.pXi = pJ\.pjr^.
But the latter is medial ; [x. 21]
therefore x . pX*, or pX* . pX* >/i - ^, is medial.
Lastly (8) p, p n/i - ^ have the remaining property in the enunciation ; therefore pX*, p\* Ji —f^ have it also. [x. 14]
(Euclid has not the assistance of symbols to prove the proportion (3) above. He therefore uses the lemmas ab\bc = a\c and (P:de = d:e to deduce from the relations
ab = (P I and d\b = c:e J
that a:c = d\e,)
The straight lines pX*, pk*Ji-'^ may take any of the following forms according as the straight lines first taken are
(i) ^, ^B, y^T?; (2) ^A, JB, JA^WA, (3) ^A, b, j'A^r^,
(I) ^a^B,
JaJB
^ »jB{A-'f^A)
(2) <IAB, ^= .
bjA-f^A
(3) >/^n/^,
JbJA
74
BOOK X
[x. 32, Lemma
II. If the other conditions are the same, but the square on the first medial straight line is to exceed the square on the second by the square on a straight line incommensurable with the first, we begin with the three straight
lines ft p sl^i —pz^-zzzz^, and the medial straight lines are
Ji'+J^
M
The possible forms are even more various in this case owing to the more various forms that the original lines may take, e.g.
(I) (*) (3) (4)
(s)
a,
JA, JA, JA,
JB, JB,
•Jt^-C
■jA-i
■Ja^ •Ja^V*
•Ja-c
the medial straight lines corresponding to these being
(0 (») (3) (4) (5)
JTljB, -JTJA, -JTTfA,
Has. Has,
'JQa '
■JbljA ' ■s/BjA-c')
Hab '
^WJA^^)
Hab '
Lemma.
Let ABC be a right-angled triangle having the angle A right, and let the perpendicular AD be drawn ;
I say that the rectangle CB, BD is equal to the square on BA, the rectangle BC, CD equal to the square on CA,
the rectangle BD, DC equal to the square on AD,
and, further, the rectangle BC, AD equal to the rectangle
Lemma, x. 33] PROPOSITIONS 32, 33 75
»
And first that the rectangle CB, BD is equal to the square on BA.
For, since in a right-angled triangle y^Z? has been drawn from the right angle perpendicular to the base, therefore the triangles ABD, ADC are similar both to the whole ABC and to one another. [vi. 8]
And since the triangle ABC is similar to the triangle ABD, therefore, as CB is to BAy so is BA to BD ; • [vi. 4]
therefore the rectangle CB, BD is equal to the square on AB.
[vi. 17] For the same reason the rectangle BC, CD is also equal to the square on AC
And since, if in a right-angled triangle a perpendicular be drawn from the right angle to the base, the perpendicular so drawn is a mean proportional between the segments of the base, [vi. 8, Por.]
therefore, as BD is to DA, so is AD to DC ;
therefore the rectangle BD, DC is equal to the square on AD.
[vi. 17]
I say that the rectangle BC, AD is also equal to the rect- angle BA, AC
For since, as we said, ABC is similar to ABD,
therefore, as BC is to CA, so is BA to AD. [vi. 4]
Therefore the rectangle BC, AD is equal to the rectangle BA, AC. [vi. 16]
Q. E. D.
Proposition 33.
To find two straight lines incommensurable in square which make the sum of the squares on them rational but the rectangle contained by them medial.
Let there be set out two rational straight lines AB, BC commensurable in square only and such that the square on the greater A Bis greater than the square on the less BC by the square on a straight line in- commensurable with AB,
[x. 30]
76 BOOK X [x. 33
«
let BC be bisected at /?,
let there be applied to AB a parallelogram equal to the square on either of the straight lines BD^ DC and deficient by a square figure, and let it be the rectangle AE^ EB ; [vi. 28]
let the semicircle AFB be described on AB,
let EF be drawn at right angles to AB,
and let AF, FB be joined.
Then, since AB, BC are unequal straight lines,
and the square on AB is greater than the square on BC by the square on a straight line incommensurable with AB,
while there has been applied X,o AB 2i parallelogram equal to the fourth part of the square on BC, that is, to the square on half of it, and deficient by a square figure, making the rect- angle AE, EB,
therefore AE is incommensurable with EB. [x. 18]
And, as -^^ is to EB, so is the rectangle BA, AE to the rectangle AB, BE,
while the rectangle BA, AE is equal to the square on AF,
and the rectangle AB, BE to the square on BF ;
therefore the square on AF is incommensurable with the square on FB ;
therefore AF, FB are incommensurable in square.
And, since AB is rational, therefore the square on A B is also rational ; so that the sum of the squares on AF, FB is also rational.
[I. 47]
And since, again, the rectangle AE, EB is equal to the square on EF,
and, by hypothesis, the rectangle AE, EB is also equal to the square on BD,
therefore FE is equal to BD ;
therefore BC is double of FE,
so that the rectangle AB, BC is also commensurable with the rectangle AB, EF
But the rectangle AB, BC is medial ; [x. 21]
therefore the rectangle AB, EF is also medial. [x. 23, Por.]
X. 33] PROPOSITION 33 77
But the rectangle ABy EF is equal to the rectangle AFy FB ; [I^mma]
therefore the rectangle AF^ FB is also medial.
But it was also proved that the sum of the squares on these straight lines is rational.
Therefore two straight lines AF, FB incommensurable in square have been found which make the sum of the squares on them rational, but the rectangle contained by them medial.
Q. E. D.
Euclid takes the straight lines found in x. 30, viz. p, . - .
VI + ^
He then solves geometrically the equations
P'
.(i).
If X, y are the values found, he takes u^ v such that
"AZ) ■; <*
and t/, V are straight lines satisfying the conditions of the problem. Solving algebraically, we get (if x>y)
whence « = -7- a / i +
y
■ ^
•(3).
Euclid's proof that these straight lines fulfil the requirements is as follows.
(a) The constants in the equations (i) satisfy the conditions of x. 18 ; therefore x ^^y.
But X :y = u^ :z^.
Therefore «' v./ 1;*,
and u, V are thus incommensurable in square,
(ft) i^ + z^ = p*, which is rational,
(r) ByOX ^^=
2 Vi+>^ By (2), uv^^p.sTo^
2n/i+/^'
78 BOOK X [x. 33. 34
But , is a medial area,
therefore uv is medial.
Since p, . ^ may have any of the three forms
(i) a, J'^^^, (2) JA, s/A^^, is) JA, Ja^, u, V may have any of the forms
, , /a + Vab /a-sJab (^) V — ; — ' V---; — '
(3)
/A + bJA lA-b^A
V i ' V ^ •
Proposition 34.
To find two straight lines incommensurable in square which make the sum of the squares on them medial but t/ie rectangle contained by them, rational
Let there be set out two medial straight lines ABy BC, commensurable in square only, such that the rectangle which they contain is rational, and the square on AB is greater than the square on BC by the square on a straight line incom- mensurable with AB ; [x. 31, ad fin,]
let the semicircle ADB be described on AB,
let BC be bisected at E,
let there be applied to AB a parallelogram equal to the square on BE and deficient by a square figure, namely the rectangle AF, FB ; [VI. 28]
therefore AFvs, incommensurable in length with FB. [x. 18]
Let FD be drawn from F at right angles to AB,
and let AD, DB be joined.
X. 34] PROPOSITIONS 33, 34 79
Since AF is incommensurable in length with FB^ therefore the rectangle BA, AF is also incommensurable with the rectangle AB^ BF. [x. n]
But the rectangle BAy AF is equal to the square on AD, and the rectangle AB, BF to the square on DB ; therefore the square on AD is also incommensurable with the square on DB.
And, since the square on AB is medial,
therefore the sum of the squares on AD, DB is also medial.
[ill. 31, 1. 47]
And, since BC is double of DF, therefore the rectangle AB, BC is also double of the rectangle AB, FD.
But the rectangle AB, BC is rational ; therefore the rectangle AB, FD is also rational. [x. 6]
But the rectangle AB, FD is equal to the rectangle AD, DB ; [Lemma]
so that the rectangle AD, DB is also rational.
Therefore two straight lines AD, DB incommensurable in square have been found which make the sum of the squares on them medial, but the rectangle contained by them rational.
Q. E. D. In this case we take [x. 31, 2nd part] the medial straight lines
P ?_„
Solve the equations
_P
Take u, v such that, if a;, ^ be the result of the solution,
.(I).
,i*
(I+>t»)*
and u, V are straight lines satisfying the given conditions.
Euclid's proof is similar to the preceding, (a) From (i) it follows [x. 18] that
whence «• ^ t^,
and n, v are thus incommensurable in square.
•(*).
8o BOOK X [x. 34, 35
(J3) t^-^z^^ —i=-==- 9 which is a medial area.
(y) uv = — ^— r . Viy
= - . -^-Ti I which is a rational area. 2 1+^'
Therefore uv is rational.
To find the actual form of «, », we have, by solving the equations (i) (if x>y\
2(l+>^)*
2(1+^)* and hence u = , ^ JQTTW+k^
« = ■
V2(l+^)
Bearing in mind the forms which — - — 7 , — 0 ^^Y take (see note
(! + >&»)* (i+>^)* on X. 31), we shall find that u, v may have any of the forms
(3) JUA^b)jA^^ JUa^^J^^
Proposition 35.
To find two straight lines incommensurable in square which make the sum of the squares on them medial and the rectangle contained by them medial and moreover incommensurable with the sum of the sqtuires on them.
Let there be set out two medial straight lines AB, BC commensurable in square only, containing a medial rectangle, and such that the square on AB is greater than the square on BC by the square on a straight line incommensurable with AB ; [x, 32, adfin.^
X. 35] PROPOSl-riONS 34, 3S 81
let the semicircles/?^ be described on AB^ and let the rest of the constructiori be as above.
Then, since AF is incommensurable in length with FB^
[X. 18] AD is also incommensurable in square with DB. [x. n]
And, since the square on AB is medial, therefore the sum of the squares on AD, DB is also medial.
[m. 31, I. 47] And, since the rectangle AF, FB is equal to the square on each of the straight lines BE, DF,
therefore BE is equal to DF\
therefore BC is double of /^Z),
so that the rectangle AB, BC is also double of the rectangle AB, FD.
But the rectangle AB, BC is. medial ;
therefore the rectangle AB, FD is also medial. [x. 32, Por.]
And it is equal to the rectangle AD, DB ;
[Lemma after x. 32]
therefore the rectangle AD, DB is also medial.
And, since AB is incommensurable in length with BC,
while CB is commensurable with BE,
therefore AB is also incommensurable in length with BE,
[X.13] so that the square on AB is also incommensurable with the rectangle AB, BE. [x. n]
But the squares on AD, DB are equal to the square on AB, [1. 47]
and the rectangle AB, FD, that is, the rectangle AD, DB, is equal to the rectangle -^^, BE',
therefore the sum of the squares on AD, DB is incommen- surable with the rectangle AD, DB.
H. E. III. 6
82 BOOK X [x. 3S
Therefore two straight lines AD, DB incommensurable in square have been found which make the sum of the squares on them medial and the rectangle contained by them medial and moreover incommensurable with the sum of the squares on them.
Q. E. D. Take the medial straight lines found in x. 32 (2nd part), viz.
Solve the equations
a: + J/ = pX* 1
,^_ZjJ^_ } <')'
(2),
and then put 1^ = pk* . x
where x, y are the ascertained values of oc, y.
Then u, v are straight lines satisfying the given conditions.
Euclid proves this as follows, (a) From (i) it follows [x. 18] that x w^.
Therefore «' >^ »■,
and u s^v,
(P) u* + t^ = p^JX, which isa medial area (3).
(y) uv = pX* . Jxy
= - -7===, which is a medial area (4)-
therefore uv is medial.
(«) '^*-i-A'
whence p' JX %^ - ^ ^ .
That is, by (3) and (4),
(«* + «/*) w uv. The actual values are found thus. Solving the equations (i), we have
pX* / ^ \
whence
M* / *~
X. 35, 36] PROPOSITIONS 35, 36 83
According as p is of the form a or J A, we have a variety of forms for tf, v^ arrived at by using the same transformations as in the notes on x. 30 and X. 32 (second part), e.g.
(2) /UaT~JC)JB^ JUA-^C)JB_ ^^^ /QA+c)J£^ JUA-c)JB_
and the expressions in (2), (3) with b in place of JB.
Proposition 36. 1/ two rational straight lines commensurable in square only be added together y the whole is irrational ; and let it be called binomial.
For let two rational straight lines AB, BC commen- 5 surable in square only be added together ;
I say that the whole -^C is ir- ^ ^ ^
rational.
For, since AB is incommensurable in length with BC — 10 for they are commensurable in square only — and, as AB is to BCy so is the rectangle AB^ BC to the square on BC^
therefore the rectangle AB, BC is incommensurable with the
square on BC [x. nj
15 But twice the rectangle ABy BC is commensurable with
the rectangle AB, BC [x. 6], and the squares on AB, BC are
commensurable with the square on BC — for AB, BC are
rational straight lines commensurable in square only — [x. 15]
therefore twice the rectangle ABy BC is incommensurable
20 with the squares on ABy BC [x. 13]
And, componendoy twice the rectangle ABy BC together
with the squares on ABy BCy that is, the square on AC [n. 4],
is incommensurable with the sum of the squares on AB, BC
[x. 16] But the sum of the squares on AB, BC is rational ;
25 therefore the square on AC is irrational,
so that AC is also irrational. [x. Def. 4]
And let it be called binomial. q. e. d.
84 BOOK X [x. 36, 37
Here begins the first hexad of propositions relating to compound irrational straight lines. The six compound irrational straight lines are formed by adding two parts, as the corresponding six in Props. 73 — 78 are formed by subtraction. The relation between the six irrational straight lines in this and the next five propositions with those described in Definitions 11. and the Props. 48 — 53 following thereon (the firsts second^ thirds fourth^ fifth and sixth binomials) will be seen when we come to Props. 54 — 59 ; but it may be stated here that the six compound irrationals in Props. 36 — 41 can be found by means of the equivalent of extracting the square root of the compound irrationals in x. 48— r53.(the process being, strictly speaking, the finding of the sides of the squares equal to the rectangles contained by the latter irrationals respectively and a rational straight line as the other side), and it is therefore the further removed compound irrational, so to speak, which is treated first.
In reproducing the proofs of the propositions, I shall for the sake of simplicity call the two parts of the compound irrational straight line oc, y^ explaining at the outset the forms which x, y really have in each case ; x will always be supposed to be the greater segment.
In this proposition jc, y are of the form p, ^k . p, and {x -^y) is proved to be irrational thus.
X *^ y, so thsit X y^ y.
Now X :y = ^ : xy,
so that a;" w xy.
But a^ ^ (a^ +y), and xy ^ ixy ; therefore (a:* +y) >./ 2xy,
and hence (a:* +y + 2xy) >./ {a^ +y).
' But {31^ +y) is rational ; therefore (x +yy, and therefore (x +y)y is irrational.
This irrational straight line, p+ jk.pfis called a binomial straight line.
This and the corresponding apotome (p-»Jk,p) found in x. 73 are the positive roots of the equation
or* - 2 ( I + >^) p» . ^ + ( I - it )« p* = o.
Proposition 37.
If two medial straight lines commensurable in square only and containing a rational rectangle be added together, the whole is irrational ; and let it be called a first bimedial straight line.
For let two medial straight lines AB, BC commensurable in square only and containing
a rational rectangle be added ^ g ^
together;
I say that the whole AC \^ irrational.
For, since AB is incommensurable in length with BCy therefore the squares on AB, BC are also incommensurable with twice the rectangle AB, BC\ [cf. x. 36, II. 9—20]
X. 37. 38] PROPOSITIONS 36—38 85
and, componendo, the squares on AB, BC together with twice the rectangle A By BC, that is, the square on AC, [n. 4], is incommensurable with the rectangle AB, BC [x. 16]
But the rectangle ABy BC is rational, for, by hypothesis, A By BC are straight lines containing a rational rectangle ; therefore the square on -^C is irrational ; therefore AC is irrational. [x Def. 4]
And let it be called a first bimedial straight line.
Q. E. D.
Here Xy^ have the forms ^py Irp respectively, as found in x. 27.
Exactly as in the last case we prove that
0^ +_)^ w 2xyy whence {x -^-yf sj ixy.
But xy is rational ; therefore (x +J')', and consequently {x +y)y is irrational.
The irrational straight line Irp + Jrp is called a first bimedial straight line.
This and the corresponding first apotome of a medial (frp - /rp) found in X. 74 are the positive roots of the equation
a^-2jk{i+J^)pKx' + k(i-kyp^ = o.
Proposition 38.
// two medial straight lines commensurable in square only and containing a medial rectangle be added together y the whole is irrational; and let it be called a second bimedial straight line.
S For let two medial straight lines ABy BC commensurable
in square only and containing
a medial rectangle be added a . P o
together; ^ h Q
I say that AC \s irrational. 10 For let a rational straight
line DE be set out, and let the
parallelogram DF equal to the
square on ^C be applied to DEy
producing DG as breadth. [i. 44]
15 Then, since the square on AC \s equal to the squares on
ABy BC and twice the rectangle ABy BCy [11. 4]
let EHy equal to the squares on ABy BCy be applied to DE\
E F
86 BOOK X [x. 38
therefore the remainder HF is equal to twice the rectangle AB, BC.
20 And, since each of the straight lines ABy BC is medial,
therefore the squares on ABy BC are also medial.
But, by hypothesis, twice the rectangle ABy BC is also medial.
And EH is equal to the squares on AB, BCy
25 while FH is equal to twice the rectangle ABy BC\
therefore each of the rectangles EH^ HF is medial.
And they are applied to the rational straight line DE ;
therefore each of the straight lines DHy HG is rational and incommensurable in length with DE. [x. 22]
30 Since then AB is incommensurable in length with BCy
and, as AB is to BCy so is the square on AB to the rectangle ABy BC,
therefore the square on AB is incommensurable with the rect- angle ABy BC. [x. 11] .
35 But the sum of the squares on ABy BC is commensurable
with the square on ABy [x. 15] ^
and twice the rectangle ABy BC is commensurable with the rectangle ABy BC [x. 6]
Therefore the sum of the squares on ABy BC is incom- 40 mensurable with twice the rectangle AB, BC [x. 13] ^
But EH is equal to the squares on ABy BCy
and HF is equal to twice the rectangle ABy BC
Therefore EH is incommensurable with HFy
so that DH is also incommensurable in length with HG.
[vi. I, X. 11]
45 Therefore DHy HG are rational straight lines commen- surable in square only ;
so that DG is irrational. [x. 36]
But DE is rational ;
and the rectangle contained by an irrational and a rational <
so straight line is irrational ; [cf. x. 20] (
therefore the area DF is irrational,
and the side of the square equal to it is irrational. [x. Def. 4]
X. 38, 39] PROPOSITIONS 38, 39 87
But AC is the side of the square equal to DF; therefore AC is irrational 55 And let it be called a second bimedial straight line.
Q. E. D.
After proving (1. 21) that eacA of the squares on AB^ BC is medial, Euclid states (IL 24, 26) that EH^ which is equal to the sum of the squares, is a medial area, but does not explain why. It is because, by hypothesis, the squares on AB^ BC are commensurable, so that the sum of the squares is commensurable with either [x. 15] and is therefore a medial area [x. 23, Por.}
In this case [x. 28, note] oc, j' are of the forms Irp^ k^pji^ respectively. Apply each of the areas {31^ -^y) and 2xy to a rational straight line <r, i.e. suppose
2xy = av.
Now it follows from the hypothesis, x. 15 and x. 23, Por. that (^+y) is a medial area ; and so is 2jcy, by hypothesis ; therefore <ru, av are medial areas.
Therefore each of the straight lines Uy v is rational and w o- (i).
Again xs^y,
therefore 3^ sj xy.
But 3^ ^ 3^ +y and 3cy ^ 2xy ; therefore ^ +y w axy, or <ru sj aVf whence u^v (2).
Therefore, by (i), {2), «, » are rational and '^.
It follows, by X. 36, that (u + v) is irrational.
Therefore {u + v)(r is an irrational area [this can be deduced from x. 20 by reducHo ad adsurdum],
whence {x +y)\ and consequently {x h-j/), is irrational.
The irrational straight line ^p-^—r is called a seamd Inmedial straight line.
This and the corresponding second apotome of a medial yrp-'^ p\ found in X. 75 are the positive roots of the equation
Proposition 39.
If two straight lines incommensurable in square which make the sum of the squares on them rational, but the rectangle contained by them medial, be culded together, the whole straight line is irrational: and let it be called major.
88 BOOK X [x. 39, 40
For let two straight lines ABl BC incommensurable in square, and fulfilling the given con- ditions [x. 3 j], be added together ; ^ g b I say that AC is irrational.
Fori since the rectangle AB, BC is medial, twice the rectangle AB, BC is also medial. [x. 6 and 23, Por.]
But the sum of the squares on AB, BC is rational ; i
therefore twice the rectangle AB, BC is incommensurable with the sum of the squares on AB, BC, so that the squares on AB, BC together with twice the rect- angle ABy BC, that is, the square on AC, is also incommen- surable with the sum of the squares on AB, BC ; [x. 16]
therefore the square on AC is irrational, so that AC IS also irrational. [x- Def. 4]
And let it be called major.
Q. E. D. Here x, y are of the fonn found in x. 33, viz.
By hypothesis, the rectangle ocy is. medial ; therefore 2xy is medial.
Also (jc* + j^ is a rational area.
Therefore x* +y w 2xy,
whence {x + j/)" w (jc" +y),
so that {x +y)\ and therefore {x +^), is irrational.
The irrational straight line -7- a./ i + j + -r- a./i . is
called a major (irrational) straight line.
This and the corresponding minor irrational found in x. 76 are the positive roots of the equation
Proposition 40.
If two straight lines incommensurable in square which make the sum of the squares on them medial, but the rectangle contc^ined by them rational, be acUled together, the whole straight line is irrational; and let it be callecl the side of a rational plus a medial area.
y
X. 40, 4i] PROPOSITIONS 39—41 89
For let two straight lines ABy BC incommensurable in square, and fulfilling the given con- ditions [x. 34], be added together ; a b o
I say that AC \s irrational.
For, since the sum of the squares on AB, BC is medial, while twice the rectangle ABj BC is rational, therefore the sum of the squares on AB, BC is incommen- surable with twice the rectangle ABy BC ; so that the square on ^C is also incommensurable with twice the rectangle AB, BC [x. 16]
But twice the rectangle ABy BC is rational ; therefore the square on AC \s irrational.
Therefore AC is irrational. [x. Def. 4]
And let it be called the side of a rational plus a medial area.
Q. E. D. Here Xy y have [x. 34] the forms
In this case {x^ +y) is a medial, and 2ocy a rational, area ; thus
jc* +y V 2xy» Therefore (x -{-yf w 2xyy
whence, since 2xy is rational,
{x +J')', and consequently (x +y)y is irrational. The irrational straight line
>/2(i+>6>) n/2(i+>^)
is called (for an obvious reason) the " side ^^ of a rationcUplus a medial (area). This and the corresponding irrational with a minus sign found in x. 77 are the positive roots of the equation
2 M^
VTh?'^ (i + i*)*'^
Proposition 41.
If two straight lines incommensurable in square which make the sum of the squares on them medial, and the rectangle contained by them medial and also incommensurable with the sum of the squares on them^ be added together , the whole straight line is irrational ; and let it be called the side of the sum of two medial areas.
90
BOOK X
[x. 41
-r
For let two straight lines AB, BC incommensurable in square and satisfying the given conditions [x. 35] be added together ; I say that -^C is irrational.
Let a rational straight line DE be set out, and let there be applied to DE the rectangle DF equal to the squares on AB, BC, and the rectangle GH equal to twice the rectangle AB,BC\
therefore the whole DHis equal to the square on AC. [n. 4]
Now, since the sum of the squares on AB, BC is medial, __
and is equal to DF, ^
therefore DF is also medial.
And it is applied to the rational straight line DE ; therefore DG is rational and incommensurable in length with DE. [x. 22]
For the same reason GK is also rational and incommen- surable in length with GF, that is, DE.
And, since the squares on AB, BC are incommensurable with twice the rectangle AB, BC, DF is incommensurable with GH ; so that DG is also incommensurable with GK. [vi. i, x. n]
And they are rational ; therefore DG, GK are rational straight lines commensurable in square only ; therefore DK is irrational and what is called binomial, [x. 36]
But DE is rational ; therefore DH is irrational, and the side of the square which is equal to it is irrational. [x. Def. 4]
But AC IS the side of the square equal to HD ; therefore -^C is irrational.
And let it be called the side of the sum of two medial areas.
In this case x, y are of the form pXi / —k-
Q. E. D.
i/'
c 1
7i
•JTT¥'
X. 41, Lemma] PROPOSITION 41 91
By hypothesis, (a:* +y^) and 2xy are medial areas, and
^+y^ 2xy (i).
* Apply ' these areas respectively to a rational straight line cr, and suppose
^+y=<^\ (a).
2xy = i7V ) Since then au and w are both medial areas, u, v are rational and both
are K^ <r (3).
Now, by (i) and (2),
OV W iJVy
so that u yjv.
By this and (3), «, v are rational and ^^. Therefore [x. 36] (u + z') is irrational. Hence <r (« + «;) is irrational [deduction from x. 20]. Thus (x +yfi and therefore (x +y), is irrational. The irrational straight line
pX* / k p\i / ~~k
is called (again for an obvious reason) the "stiW of the sum of two medials (medial areas).
This and the corresponding irrational with a minus sign found in x. 78 are the positive roots of the equation
:»:*- 2 V^. a;»p» + X j-j-^p*= o.
Lemma.
And that the aforesaid irrational straight lines are divided only in one way into the straight lines of which they are the sum and which produce the types in question, we will now prove after premising the following l^mma.
Let the straight line AB be set out, let the whole be cut into unequal parts at each of
the points C, /?, .
and let-^Cbe supposed greater ^ dec b
than DB ;
I say that the squares on AC, CB are greater than the squares on AD, DB.
For let AB be bisected at E.
Then, since -^C is greater than DB, let DC be subtracted from each ; therefore the remainder AD is greater than the remainder CB.
But AE is equal to EB ; therefore DE is less than EC ;
92 BOOK X [Lemma, x. 42
therefore the points C, D are not equidistant from the point of bisection.
And, since the rectangle AC, CB together with the square on EC is equal to the square on EB, [n. 5]
and, further, the rectangle AD, DB together with the square on DE is equal to the square on EB, \id^
therefore the rectangle AC, CB together with the square on EC is equal to the rectangle AD, DB together with the square on DE.
And of these the square on DE is less than the square on EC\
therefore the remainder, the rectangle AC, CB, is also less than the rectangle AD, DB,
so that twice the rectangle AC, CB is also less than twice the rectangle AD, DB.
Therefore also the remainder, the sum of the squares on AC, CB, is greater than the sum of the squares on AD, DB.
Q. E, D.
3. and which produce the types in question. The Greek is irotov0^i!v rd rpoKel/uva aSfl, and I have taken dSri to mean "types (of irrational straight lines)," though the expression might perhaps mean " satisfying the conditions in question/'
This proves that, if x -^-y^u + v, and if u, v are more nearly equal than X, y (i.e. if the straight line is divided in the second case nearer to the point of bisection), then
(:t»+y)>(i/> + «;»).
It is first proved by means of 11. 5 that
2Xy<2UV,
whence, since {x -^yY = (» + vf^ the required result follows.
Proposition 42.
A binomial straight line is divided into its terms at one point only.
Een?AftJi^a binomial straiefht line divided into its terms atC; ^^— ^
therefore AC, CB are rational ^f;^^ - — -^^ g b
straight lines commensurable in
square only. jT.x. 36]
I say that AB is not divided at another point into two rational straight lines commensurable in square only.
X. 42] PROPOSITION 42 93
For, if possible, let it be divided at D also, so that AD, DB are also rational straight lines commensurable in square only.
It is then manifest that AC\^ not the same with DB.
For, if possible, let it be so.
Then AD will also be the same as CBy
and, as .^C is to CB, so will BD be to DA \
thus AB will be divided at D also in the same way as by the division at C :
which is contrary to the hypothesis.
Therefore AC\^ not the same with DB.
For this reason also the points C, D are not equidistant from the point of bisection.
Therefore that by which the squares on AC, CB differ from the squares on AD, DB is also that by which twice the rectangle AD, DB differs from twice the rectangle
AC, CB,
because both the squares on AC, CB together with twice the rectangle AC, CB, and the squares on AD, DB together with twice the rectangle AD, DB, are equal to the square on AB. [11. 4l
But the squares on AC, CB differ from the squares on
AD, DB by a rational area,
for both are rational ;
therefore twice the rectangle AD, DB also differs from twice the rectangle AC, CB by a rational area, though they are medial [x. 21] :
which is absurd, for a medial area does not exceed a medial by a rational area. [x. 26]
Therefore a binomial straight line is not divided at different points ;
therefore it is divided at one point only.
Q. E. D.
This proposition proves the equivalent of the well-known theorem in surds that,
then a=-x, b=y,
and if Ja + Jb = ^x-\- Jy,
then a=^x, b=y (or a=y, b = x).
1
94 BOOK X [x. 42, 43
The proposition states that a binomial straight line cannot be split up into terms {ovo/jLara) in two ways. For, if possible, let
x+y = x' +y, where x, y, and also x\ y\ are the terms of a binomial straight line, x\ y' being different from a;, y (or y^ x).
One pair is necessarily more nearly equal than the other. Let x\y be more nearly equal than x, y.
Then (x" +/) - (pc^ +/*) = 2x'y* - 2xy,
Now by hypothesis (x* +y), (jc'' +y*) are rational areas, being of the form
but 2x'y\ 2xy are medial areas, being of the form »jk,^\ therefore the difference of two medial areas is rational :
which is impossible. [x. 26]
Therefore o^^y' cannot be different from x^y (or^*, x\
Proposition 43.
A first bimedial straight line is divided at one point only.
Let AB be a first bimedial straight line divided at C, so ;;
that AC, CB are medial straight *
lines commensurable in square . — . =
only and containing a rational
rectangle ; [x. 37]
I say that AB is not so divided at another point.
For, if possible, let it be divided at D also, so that AD, DB are also medial straight lines commensurable in square only and containing a rational rectangle.
Since, then, that by which twice the rectangle AD, DB differs from twice the rectangle AC, CB is that by which the squares on AC, CB differ from the squares on AD, DB,
while twice the rectangle AD, DB differs from twice the rectangle AC, CB by a rational area — for both are rational —
therefore the squares on AC, CB also differ from the squares on AD, DB by a rational area, though they are medial :
which is absurd. [x. 26]
Therefore a first bimedial straight line is not divided into \
its terms at different points ; (
therefore it is so divided at one point only. \
Q. E. D.
X. 43. 44]
PROPOSITIONS 42—44
95
In this case, with the same hypothesis, viz. that
and X , y are more nearly equal than x, j,
we have as before (re* +y) - (jc'* +y*) = 2xy - 2xy.
But, from the given properties of x, y, and x\ y\ it follows that locy^ 2x'y are rational^ and (a:*+y), (^'*+y*) nudiai^ areas.
Therefore the difference between two medial areas is rational : which is impossible. [x. 26]
Proposition 44. A second bimedial straight line is divided at one point only.
Let AB be a second bimedial straight line divided at C, so that AC, CB are medial straight lines commensurable in square only and containing a medial rectangle ; [x. 38]
It is then manifest that C is not at the point of bisection, because the segments are not commensurabld* in length.
I say that AB is not so divided at another point
A DOB
D
— h-
o
M
F L Q K
For, if possible, let it be divided at D also, so that AC is not the same with DB, but AC is supposed greater ; it is then clear that the squares on AD, DB are also, as we proved above [Lemma], less than the squares on AC, CB ; and suppose that AD, DB are medial straight lines commen- surable in square only and containing a medial rectangle.
Now let a rational straight line EF be set out, let there be applied to EF the rectangular parallelogram EK equal to the square on AB,
and let EG equal to the squares on AC, CB be subtracted ; therefore the remainder HK is equal to twice the rectangle AC, CB. • [II. 4]
Again, let there be subtracted EL, equal to the squares on AD, DB, which were proved less than the squares on AC, CB [Lemma] ;
96 BOOK X [x. 44
therefore the remainder MK is also equal to twice the rect- angle AD, DB.
Now, since the squares on AC, CB are medial, therefore EG is medial.
And it is applied to the rational straight line EF ;
therefore EH is rational and incommensurable in length with EF. [x. 22]
For the same reason
HN is also rational and incommensurable in length with EF,
And, since AC, CB are medial straight lines commen- surable in square only, therefore AC is incommensurable in length with CB.
But, as y^C is to CB, so is the square on ^C to the rect- angle AC, CB ;
therefore the square on-^C is incommensurable with the rect- angle AC, CB. [x. ^i]
But the squares on AC, CB are commensurable with the j
square on AC ; for AC, CB are commensurable in square. J
[X.15]
And twice the rectangle AC, CB is commensurable with ^
the rectangle AC, CB. [x. 6]
Therefore the squares on AC, CB are also 'incommen- surable with twice the rectangle AC, CB. [x. 13]
But EG is equal to the squares on A C, CB, and I/K is equal to twice the rectangle AC, CB ; therefore EG is incommensurable with I/K, so that E/f is also incommensurable in length with NN.
[vi. I, X. 11]
And they are rational ; therefore E/f, HN are rational straight lines commensurable in square only.
But, if two rational straight lines commensurable in square only be added together, the whole is the irrational which is called binomial. [x. 36]
Therefore EN is a binomial straight line divided zi