the words ix' subes, that is, in a ftraight line, or in the fame Book I. direction, be plain, when two ftraight lines are faid to be in a ftraight line, it does not appear what ought to be understood by these words, when a ftraight line and a curve, or two curve lines, are faid to be in the fame direction; at least it cannot be explained in this place; which makes it probable that this definition, and that of the angle of a fegment, and what is faid of the angle of a femicircle, and the angles of fegments, in the 16. and 31. propofitions of book 3. are the additions of fome lefs fkilful editor: On which account, especially fince they are quite useless, these definitions are distinguished from the rest by inverted double commas. DEF. XVII. B. I. The words," which alfo divides the circle into two equal "parts," are added at the end of this definition in all the copies, but are now left out as not belonging to the definition, being only a corollary from it. Proclus demonftrates it by conceiving one of the parts into which the diameter divides the circle, to be applied to the other; for it is plain they muft coincide, elfe the ftraight lines from the centre to the circumference would not be all equal: The fame thing is easily deduced from the 3. prop. of book 3. and the 24. of the fame; from the first of which it follows that femicircles are fimilar fegments of a Circle: And from the other, that they are equal to one another. DE F. XXXIII. B. I. This definition has one condition more than is necessary; because every quadrilateral figure which has its oppofite fides equal to one another, has likewife its oppofite angles equal; and on the contrary. A Let ABCD be a quadrilateral figure of which the oppofite fides AB, CD are equal to one another; as alfo AD and BC: Join BD; the two fides AD, DB are equal to the two CB, BD, and the bafe AB is equal to the base CD; there fore by prop. 8. of book 1. the angle ADB is equal to the angle CBD; and by prop. 4. B. 1. the angle BAD is equal to the angle DCB, and ABD to BDC; and therefore also the angle ADC is equal to the angle ABC. B C And Book f. And if the angle BAD be equal to the oppofite angle BCD, w and the angle ABC to ADC; the oppofite fides are equal: Because, by prop. 32. B. 1. all the angles of the quadrilateral figure ABCD are together equal to four right angles, and the two angles BAD, ADC are together equal to A B D C the two angles BCD, ABC: Where- PROP. VII. B. I. There are two cafes of this propofition, one of which is not in the Greek text, but is as neceffary as the other: And that the cafe left out has been formerly in the text appears plainly from this, that the fecond part of prop. 5. which is neceffary to the demonftration of this cafe, can be of no use at all in the elements, or any where elfe, but in this demonstration; because the fecond part of prop. 5. clearly follows from the first part, and prop. 13. B. 1. This part muft therefore have been added to prop. 5. upon account of fome propofition betwixt the 5. and 13. but none of these ftand in need of it except the 7. propofition, on account of which it has been added: Befides, the tranflation from the Arabic has this cafe explicitly demonftrated: And Proclus acknowledges that the fecond part of prop. 5. was added upon account of prop. 7. but gives a ridiculous reafon for it, "that it might afford an answer to objections made against the 7." as if the cafe of the 7. which is left out, were, as he exprefsly makes it, an objection against the propofition itself. Whoever is curious may read what Proclus fays of this in his commentary on the 5. and 7. propofitions; for it is not worth while to relate his trifles at full length. It was thought proper to change the enunciation of this 7. prop. fo as to preferve the very fame meaning; the literal tranflation from the Greek being extremely harsh, and difficult to be understood by beginners. РВО Р. PROP. XI. B. I. A corollary is added to this propofition, which is neceffary to Prop. 1. b. 11. and otherwife. PROP. XX. and XXI. B. I. Proclus, in his commentary, relates, that the Epicureans derided this propofition, as being manifeft even to affes, and needing no demonftration; and his anfwer is, that though the truth of it be manifeft to our fenfes, yet it is fcience which must give the reason why two fides of a triangle are greater than the third: But the right answer to this objection against this and the 21ft, and fome other plain propofitions, is, that the number of axioms ought not to be increased without neceffity, as it must be if these propofitions be not demonftrated. Monf. Clairault, in the preface to his elements of geometry, published in French at Paris, ann. 1741, fays, That Euclid has been at the pains to prove, that the two fides of a triangle which is included within another are together lefs than the two fides of the triangle which includes it; but he has forgot to add this condition, viz. that the triangles must be upon the fame base; because, unless this be added, the fides of the included triangle may be greater than the fides of the triangle which includes it, in any ratio which is lefs than that of two to one, as Pappus Alexandrinus has demonftrated in Prop. 3. b. 3. of his mathematical collections. PROP. XXII. B. I. Some authors blame Euclid becaufe he does not demonftrate, that the two circles made ufe of in the conftruction of this problem muft cut one another: But this is very plain from the determination he has given, viz. that any two of the ftraight lines DF, FG, GH mufl be great er than the third: For who is fo F G H that, for the like reafon, the circle defcribed from the centre G, at the distance GH or GM, muft meet DG betwixt D a 30x Book I. Book 1. and G; and that thefe circles must meet one another, because FD and GH are together greater than FG? And this determination is easier to be understood than that which Mr Thomas Simpson derives from it, and puts inftead of Euclid's, in the 49th page of his elements of geometry, that he D M may fupply the omiffion he blames F G H Euclid for; which determination is, that any of the three ftraight lines must be less than the fum, but greater than the difference of the other two: From this he fhews the circles must meet one another, in one cafe; and fays, that it may be proved after the fame manner in any other cafe: But the straight line GM which he bids take from GF may be greater than it, as in the figure here annexed; in which cafe his demonftration must be changed into another. PROP. XXIV. B. I. D To this is added, " of the two fides DE, DF, let DE be Mr Thomas Simpson, in p. 262. of G F as it is very eafy to perceive, that DG being equal to DF, the point G is in the circumference of a circle defcribed from the centre D at the distance DF, and must be in that part of it which is above the ftraight line EF, because DG falls above DF, the angle EDG being greater than the angle EDF. PROP. XXIX. B. I. The propofition which is ufually called the 5th poftulate, or 11th axiom, by fome the 12th, on which this 29th depends, has given a great deal to do, both to antient and modern geome- Book I. ters: It seems not to be properly placed among the Axioms, as, indeed, it is not felf-evident; but it may be demonstrated The distance of a point from a ftraight line, is the perpendicular drawn to it from the point. DE F. 2. One ftraight line is faid to go nearer to, or further from, another ftraight line, when the diftances of the points of the first from the other straight line become lefs or greater than they were; and two ftraight lines are faid to keep the same distance from one another, when the distance of the points of one of them from the other is always the fame. AXIO M. A ftraight line cannot first come nearer to one another straight line, and then go further from it, before it cuts it; and, in like manner, a ftraight line cannot D go further from another ftraight A B E line, and then come nearer to F H it; nor can a ftraight line keep the fame diftance from another ftraight line, and then come nearer to it, or go further from it; for a straight line keeps always the fame direction. For example, the ftraight line ABC cannot first come near er to the ftraight line DE, as B from the point A to the point A C See the fi B, and then, from the point B to the point C, go further from the fame DE: And, in like man- F gure above. ner, the ftraight line FGH can not go further from DE, as from F to G, and then, from G to H, come nearer to the fame DE: And fo in the last cafe, as in fig. 2. PROP. I. If two equal ftraight lines AC, BD, be each at right angles to the fame ftraight line AB; if the points C, D be joined by the ftraight line CD, the ftraight line EF drawn from any point E in AB unto CD, at right angles to AB, fhall be equal to AC, ⚫ or BD. If EF be not equal to AC, one of them must be greater than the other; let AC be the greater; then, because FE is lefs |
