The history of philosophy, in eight parts by Thomas Stanley.

Sect. 1. Propositions invented by him.

THat he improved (as Proclus implies) the Geometry which he learnt of the Aegyptians with many propositions of his own, is confirmed by Laertius, who saith, that he much advanced those things, the invention whereof Callimachus in his Iambicks, as∣scribes to Euphorbus the Phrygian, as scalenous triangles, and others. Nor is it to be doubted, but that many of them are of those, which Euclid hath reduced into his Elements; whose design it was to collect and digest those that were invented by others, accurately demonstrating such as were more negligently pro∣ved, but of them only, these are known to be his.

[1. Every Diameter divides its circle into two equall parts.* 1.1] This proposition which Euclid makes part of the definition of a Dia∣meter,* 1.2 Proclus affirmes to have been first demonstrated by Thales.

2. [* 1.3 In all Isosceles triangles, the angles at the base are equall the one to the other, and those right lines being produced, the angles under the base are equall.]* 1.4 Proclus saith, that for the invention of this like∣wise, as of many other propositions, we are beholding to Thales, for he first observed and said, that of every Isosceles, the angels at the base are equall, and according to the antients called equall like. These are three passages in the demonstration, which infer nothing toward the conclusion, of which kind there are many in Euclid, and seem to confirm the antiquity thereof, and that it was lesse curiously reformed by him.

3. [If two lines cut one the other, the verticle angles shall equall the one the other.]* 1.5 Eudemus attests this theorem to have been invented by Thales,* 1.6 but first demonstrated by Euclid.* 1.7

4. [* 1.8 If two triangles have two angles equall to two angles the one to the other, and one side equall to one side, either that which is adjacent to the equall angles, or that which subtendeth one of the equall angles, they shall likewise have the other sides, equall to the other sides, both to both, and the remaining angle equall to the remaining angle] * Eudemus

attributes this theorem (saith Proclus) to Thales, for showing the distance of ships upon the Sea, in that manner as he is said to do, it is necessary that he perform it by this.

Pamphila (saith Laertius) affirmes, that he first described the rectangle triangle of a circle.]* 1.9 Ramus attributes to Thales (upon this authority of Laertius) the second, third, fourth, and fifth pro∣positions of the fourth book of Euclid, which are concerning the adscription of a triangle and a circle, and consequently takes 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉 here to include both inscription, and circumscrip∣tion; whereas in all those propositions, there is nothing proper to a rectangle triangle; so that if the word 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉 be retain'd, it must relate to the 31 proposition of the third book, whence may be deduced the description of a rectangle triangle in a circle. But because there is no such proposition in Euclid, and this hath but an obscure reference to part of that theorem; it is to be doubted that the Text of Laertius is corrupt, and the word (or mark) 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉 insered by accident, without which these words 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉 exactly correspond with those of* 1.10 Vitru••••ius, ••ythagoricum trigonum orthogonium descri∣bere: by which he means (as he at large expresseth* 1.11 elsewhere,) the forty fifth proposition of the first book of Euclid, that in rect∣angle triangles, the square of the hypotenuse, is equall to the square of the sides containing the right angle. That Vitruvius, Proclus, and others, attribute this invention to Pythagoras, confirmes it to be the same here meant by Laertius; who addes, that Thales, for the invention hereof, sacrificed an Oxe, though others (saith he) among whom is Apollodorus, ascribe it to Pythagoras. And in the life of Pythagoras, he cites the same Apollodorus, that Pythagoras sacrificed a Hecatomb, having foundout, that the hypotenuse of a right angled triangle, is of equall power to the two sides, including the right angle according to the Epigram

That noble scheme Pythagoras devis'd, For which a Hecatomb he sacrific'd.

Cicero, though he differ in the Author, agrees in the quantity of the offering with Laertius; affirming, that Pythagoras upon any new invention, used to sacrifice an Ox: Which kind of gratitude begun by Thales, was imitated by others also, as by Perseus.

* 1.12 Finding three spirall lines, in sections five, Perseus an offering to the Gods did give.

Notes

• * 1.1

  Lib. 1. def. 17.
• * 1.2

  in Euclid lib. 2. com. 14.
• * 1.3

  Euclid. lib. 1. prop. 5.
• * 1.4

  lib. 3. com. 9.
• * 1.5

  Euclid. lib. 1. prop. 15.
• * 1.6

  Procl. lib. 3. com. 19.
• * 1.7

  Euclid lib. 1. prop. 26.
• * 1.8

  Procl. lib. 3. com. 31.
• * 1.9

  Geom.
• * 1.10

  Lib. 10. cap. 11.
• * 1.11

  Lib. 9. cap. 2.
• * 1.12

  Procl in Euclid. lib. 2. def. 4. where the words perhaps are inverted, and for 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉.