Euclid,

ElementsI 47 (the so-called Pythagorean Theorem)©

translated by Henry Mendell (Cal. State U., L.A.)

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Proposition 47: In right-angled triangles the square from the side subtending the right angle is equal to the squares from the sides containing the right angle.

(diagram 1) Let there be a right-angled triangle ABG having as right the angle enclosed by BAG. I say that the square from BG is equal to the squares from BA, AG.

(diagram 2) Let there be written up square BDEG from BG, and HB, QG from BA, AG, (diagram 3) and through A let a parallel AL to either of BD, GE be drawn. (diagram 4) And since each of the angles by BAG, BAH is right, in fact, to some straight line, BA and to point A on it two straight-lines, AG, AH, which are not lying on the same sides, make two successive angles equal to two right angles. Therefore, GA is in a straight-line to AH. For the same reasons, in fact, BA is also in a straight-line with AQ. (diagram 5) And since the angle by DBG is equal to that by ZBA, since each is right, let a common, that by ABG, be added. Therefore, a whole, that by DBA, is equal to a whole, that by ZBG. (diagram 6) And since DB is equal to BG, while ZB is equal to BA, in fact, two, DB, BA are respectively equal to two, ZB, BG. And an angle, that DBA, is equal to an angle, that by ZBG. Therefore, a base, AD, is equal to a base, ZG, and triangle ABD is equal to triangle ZBG. (diagram 7) And parallelogram BL is double triangle ABD, since they have a base that's the same, BD, and are in the same parallels, BD, AD, (diagram 8) while square HB is double triangle ZBG, since again they have a base that's the same, ZB, and are in the same parallels, ZB, HG. (diagram 9) But doubles of equals are equal to one another. Therefore, parallelogram BL is equal to square HB. (diagram 10)(diagram 11)(diagram 12)(diagram 13)(diagram 14) (diagram 15) Similarly, in fact, with AE, BK joined, it will be also shown that parallelogram GL is equal to square QG. Therefore, a whole, square BDEG is equal to two squares, HB, QG. And while square BDEG is written up from BG, HB, QG is also from BA, AG. Therefore, the square from side BG is equal to the squares from sides BA, AG. Therefore, in right-angled triangles the square from the side subtending the right angle is equal to the squares from the sides containing the right angle, just what it was required to show.

Notes: The reputation of Pythagoras seems to depend on two lines of anaphoric elegiac poetry by one Apollodorus the arithmetician or calculator, which is quoted frequently in antiquity.

When Pythagoras found the far-famed line

(he was?) famed for which he led the famed oxen-sacrifice.

In his edition of the Diogenes Laertius, H.S. Long suggests changing the first word from 'when (hênika)' to 'he achieved (enuke)', which alters slightly the rest of the couplet:

Pythagoras achieved far-fame; he found the line,

famed for which he led the famed oxen-sacrifice.

Who was Apollodorus and what he knew of the history of mathematics is beyond conjecture other than that he lived before Cicero quoted him and that his . With nothing more than the principle that anyone with the same name mentioned by Diogenes Laertius as attributing things to Pythagoreans, von Arnim (Pauly-Wisowa, "Apollodorus (68)" thought that he might be a Apollodorus of Cyzicus who claimed that Democritus lived with Philolaus (D.L. VII 38), but we don't know anything about this Apollodorus either.^{1} That said, it is highly plausible that he attributed the discovery of I 47 to Pythagoras; otherwise, why would it be this theorem rather than any other that people mention? Plutarch's suggestion that he attributed the the application of areas is implausible, simply because no one else suggests it, while Plutarch is looking for something better to attribute, looks three theorems back in the *Elements*, and generalizes it to something yet more amazing than it is.

1. There another known Apollodorus from Cyzicus. Plato has Socrates mention a general hired by Athens in Ion 541C7 (Pauly-Wisowa, Apollodorus (25), cf. Claudius Aelian, *Varia Historia* 14.5.5-6, who is clearly just using Plato).

Here then is the evidence (translated below). Cicero mentions the sacrifice, and Vitruvius the sacrifice and the rule with for the 3, 4, 5 foot triangle (1st cent. BCE). Athenaeus, Diogenes Laertius, and Plutarch (2nd century) quote Apollodorus in mentioning both the theorem and the sacrifice, with Plutarch preferring with no evidence the application theorem (I 44). The most plausible story is that Apollodorus wrote a poem that became popular where he described the sacrifice and the rule that 3^{2} + 4^{2} = 5^{2}. This was then taken to be a claim that Pythagoras had discovered or even proved the theorem. Proclus was aware of how tenuous all this was and so described it. This has nothing to do with the question of Apollodorus' veracity, but it would be strange not to take at least a sceptical view of the evidence.

In fact, the Pythagorean relation occurs in Old Babylonian procedures, at least 1300 years before Pythagoras, while it is unlikely that Pythagoras provided a Greek geometrical demonstration of anything. It is also unlikely that Euclid was the first to prove I 47 or VI 31. It is useful to point out also that Pythagoras was not the first to find a rule for finding Pythagorean triples, numbers such that *n*^{2} + *m*^{2} = *p*^{2}. The Old Babylonian tablet, Plimpton 322, exhibits evidence for some such rule.

Of course, as Cicero points out, the story is incompatible with the view that Pythagoras was a vegetarian, but then so are many other stories told about him.

Cf. C. Huffman, "Pythagoras" in the *Stanford Encyclopedia of Philosophy* (on-line). For a different view of the evidence, cf. L. Zhmud, "Pythagoras as a Mathematician," Historia Mathematica 16 (1989): 249-68.

Here follows the texts.

Proclus, our most learned source on the history of Greek mathematics, does not actually suggest that Pythagoras proved it (*Commentary on Euclid's Elements I*, 426.6-14):

Listening to those who profess/wish (boulomenôn) to relate antiquities it is possible to find them referring this theorem back to Pythagoras and calling him 'ox-sacrificer' (or: saying that he sacrificed oxen) upon the discovery, but while I also wonder at those who first attended to the truth of this theorem, I admire more at theElementsauthor, not only since he established this through a most splendid demonstration, but that in the sixth book he also precisely determined the more universal version of it with irrefutable arguments of the science.

Proclus is cagey about whether he thinks Pythagoras discovered the theorem ('those who profess to relate' and 'it is possible to find then saying'). Furthermore, depending on what he means by 'attend to the truth', he need not suggest that everyone who attended to the truth of the theorem, including Pythagoras, actually proved it. To attend to the truth of some claim, it might be enough to see that it is true. Does Proclus think that Euclid was the first to prove I 47 or the first to provide this splendid demonstration and its generalization for similar figures (VI 31)?

Plutarch, *It not being possible to live in the manner of Epicurus* (1094A-C3, with context):

Eudoxus prayed that after standing next to the sun and learning the shape of the star and its size and its form he would burn up, as Phaethon. and Pythagoras for the diagram (proof) sacrificed an ox, as Apollodorus says:

When Pythagoras found the far-famed linewhether concerning the subtending side how it equals in part those enclosing the right-angle or whether a problem about the region of the the application (i.e. the figure applied to a line, though it could mean 'the area of a parabola'!). And the servants used to anoint Archimedes, dragging him by force from the diagrams (proofs), and he would describe the shapes on his stomach with the oil scraper, and while bathing, as they say, from overflow understood the measurement of the crown, as from some possession or inspiration he leapt out screaming, "I've found it." and he went on shouting this repeatedly.

(he was?) famed for which he led the famed oxen-sacrifice.

Plutarch, Quaest. conviv. 720A

After these words they considered it worthy for me also to contribute something to their discussion. I praised the opinions stated as true born and proper to those very people and I said that they would be adequately likely. "So that you don't frown on your own," I said, "nor look in any way outside, listen to the story that is highly esteemed among our professors on this. For there is among the geometrical theorem, or rather problems, given two forms to apply a third equal to one and similar to the other. Upon this being discovered, they also say that Pythagoras performed a sacrifice. For this is, at any rate, much more refined and of the Muses than the theorem which demonstrated the hypotenuse being in power equal to those about the right-angle." "You speak well", said Diogenianus, "but what does this have to do with the discussion?" "You will readily know," I said, "when you recall the division in theTimaeus(48e ff.), by which he divided the first things in three, by which the world got its coming to be, of which we call one by the more correct of names 'god', another 'matter', the the other 'form'.

Diogenes Laertius, *Life of Pythagoras* VIII 12

Timaeus says in the tenth book of

Historiesthat he said that the women who live with men have names of gods, Korai (girls, Persephone), Nymphs (Virgins), then Mothers. He also says that this-man brought geometry to its limit, although Moiris was the first to discover the principles of its elements. as says Anticleides in the secondOn Alexander. Most of all that Pythagoras studied the arithmetic form of it and discovered canon from one string. And Apollodorus the calculator says that he sacrificed a hundred-oxen after discovering the the subtending side equals-in-power the enclosing sides. There is also an epigram which goes thus:When Pythagoras found the far-famed line

(he was?) famed for which he led the famed oxen-sacrifice

In the Greek Anthology VII 119.1-2, this appear as:

Of Diogenes Laertius

Pythagoras achieved far-fame; he found the line,

famed for which he led the famed oxen-sacrifice.

But also see Diogenes Laertius, *Life of Thales* I 24.10-25.4

Pamphila says that having learned to do geometry among the Egyptians he was the first to describe-down (draw a diagram of) the right-angled triangle of a circle and that he sacrificed an ox, but others say Pythagoras, among whom is Apollodorus the calculator. This man advanced in the greatest amount what Callimachus says in theIambicsEuphorbus the Phrygian discovered, i.e., scalenes and triangles and whatever holds of the linear study.

Athenaeus, *Deipnosophistae* X 15 (cf. the epitome also, 2.2.27.19-22)

Hecataeus says that Egyptians are bread eaters, devouring cyllestias (spelt bread?), while grinding up barley for drink. For these reasons Alexis in the book

On Self-Rulesaid that the Bocchoris and his father Neochabis (the first a pharaoh from the 8th cent. BCE) used moderate food. And Pythagoras used moderate food, as Lycon of Iasus in the Life of Pythagoras relates. He did not hold off from animals, as Aristoxenus stated. But Apollodorus the arithmetician says that he also sacrificed a hundred-oxen on the discovery that the side subtending the right angle of a right-angled triangle equals in power the containing sides:When Pythagoras found the far-famed line

(he was?) famed for which he led the famed oxen-sacrificePythagoras was also a light-drinker and lived his life most frugally.

Cicero, On the nature of the gods III c. 36 §88

When Pythagoras discovered something new in geometry he is said to have sacrificed an ox to the Muses. But I also don't believe this, since that man was not willing even to sacrifice a victim to Apollo of Delos, lest he sprinkle the alter with blood.

Vitruvius, De architectura 9 pr. 6-7

topLikewise, Pythagoras showed how a carpenter's square might be found without ingenious constructions, and the square that carpenters by working with great labor were barely able to produce accurately, it is set out with calculations and methods from his precepts. For if three rulers are taken of which one is 3 feet, another 4 feet, the third 5 feet, and the rulers are positioned with one another so as to touch one another at their end points, having the shape of a triangle, they will form a correct carpenter's square. However, if single squares are applied with equal widths to the lengths of the individual rulers themselves, what will be three foot side will have 9 feet of area, what will be 4, 16 feet, what will be 5, 25. Then, as much as is the number of feet in area that the two squares from three foot lengths of the sides and the four make, so many will also be equal the number that one described from five feet.

When Pythagoras discovered it, with no doubt that he was taught by the Muses in that discovery, giving the greatest thanks he is said to have sacrificed victims. For the same calculation, for instance, is useful in many things and measurements, for example, it is procured in the constructions of stairs in structures where the levelings of steps get regulated.