## Girolamo Saccheri's Euclides Vindicatus |

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### Common terms and phrases

ac propterea acute angle acutus aequales aforesaid aliquo puncto altera atque aut anguli autem common perpendicular constat continuatione cujus curva curve demonstrated Demonstratur Dico distantiam duae duas duobus rectis eadem enim eodem plano equal erit erunt etiam Euclid ex praecedente ex puncto finitam finite fore four angles greater hinc hujus hypothesi anguli acuti hypothesi anguli obtusi hypothesis of acute hypothesis of obtuse hypothesis of right ideo Igitur illa infinite infinitum inter invicem ipsa ipsius Itaque join KHLK let fall linea recta loco demonstrandum major minor nempe nimirum obtuse angle Parallel Postulate plane posse postulate potest praedicta primi PROPOSITIO PROPOSITION protracta prout punctum puta quadrilateral quae quam Quare quia quidem Quod erat demonstrandum rectae lineae rectos right angle rursum Saccheri SCHOLION sint sive straight line tandem triangle trianguli vero Wherefore

### Popular passages

Page x - Ask what I shall give thee. And Solomon said, Thou hast shewed unto thy servant David my father great mercy, according as he walked before thee in truth, and in righteousness, and in uprightness of heart with thee; and thou hast kept for him this great kindness, that thou hast given him a son to sit on his...

Page x - This book has been for nearly twenty-two centuries the encouragement and guide of that scientific thought which is one thing with the progress of man from a worse to a better state. The encouragement ; for it contained a body of knowledge that was really known and could be relied on. The guide ; for the aim of every student of every subject was to bring his knowledge of that subject into a form as perfect as that which geometry had attained.

Page xi - It is certain that from its completeness, uniformity and faultlessness, from its arrangement and progressive character, and from the universal adoption of the completest and best line of argument, Euclid's " Elements " stand preeminently at the head of all human productions.

Page xviii - Saccheri lays down the clear distinction between what he calls definitiones quid nominis or nominales, and definitiones quid rei or reales, namely, that the former are only intended to explain the meaning that is to be attached to a given term, whereas the latter, besides declaring the meaning of a word, affirm at the same time the existence of the thing defined or, in geometry, the possibility of constructing it. The definitio quid nominis becomes a definitio quid rei "by means of a postulate, or...

Page xxiv - Since, then, the least numbers of those which have the same ratio measure those which have the same ratio the same number of times, the greater the greater and the less the less, that is, the antecedent the antecedent and the consequent the consequent, [vn.

Page x - Wallace says, speaking of all time before the seventeenth century: "Then going backward, we can find nothing of the first rank except Euclid's wonderful system of geometry, perhaps the most remarkable product of the earliest civilizations." Says Professor Alfred Baker, of the University of Toronto: " Of the perfection of Euclid (BC 290) as a scientific treatise, of the marvel that such a work could have been produced two thousand years ago, I shall not here delay to speak. I content myself with making...

Page xviii - The definitio quid nominis becomes a definitio quid rei " by means of a postulate, or when we come to the question whether the thing exists and it is answered affirmatively1." Definitiones quid nominis are in themselves quite arbitrary, and neither require nor are capable of proof ; they are merely provisional and are only intended to be turned as quickly as possible into definitiones quid rei, either (i) by means of a postulate in which it is asserted or...

Page xii - Euclid (BC 285): there is hardly anything in mathematics more beautiful than his wondrous fifth book; and he has also in the seventh, eighth, ninth and tenth books fully and ably developed the first principles of the Theory of Numbers, including the theory of incommensurables. We have next Apollonius (about...

Page 29 - ... angles with these perpendiculars; therefore there are three hypotheses to be distinguished according to the species of these angles. And the first indeed I will call hypothesis of right angle ; the second however, and the third I will call hypothesis of obtuse angle, and hypothesis of acute angle.

Page 21 - III. // two equal straights [sects] (fig. 3) AC, BD, stand perpendicular to any straight AB: I say the join CD will be equal to, or less, or greater than AB, according as the angles at CD are right, or obtuse, or acute.