a straight line, or in the fame direction, be plain, when two straight Book f. lines are faid to be in a straight line, it does not appear what ought to be understood by these words, when a straight 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 segment, 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 ufelefs, thefe Definitions are diftinguished from the reft by inverted double commas. DE F. 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 must coincide, else the straight lines from the center to the circumference would not be all equal. the fame thing is eafily deduced from the 31. 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 neceffary; be cause 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 opposite fides AB, CD are equal to one another; as álfo AD and BC. join BD; the two fides AD, DB are cqual to the two CB, BD, and the bafe AB is equal to the bafe CD; therefore by Prop. 8. B C 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 alfo the angle ADC is equal to the angle ABC, Book I. A D And if the angle BAD be equal to the oppofite angle BCD, 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 the two angles BCD, ABC. wherefore BAD, ADC are the half of all the four angles; that is, BAD and ADC are equal to two right angles. and therefore AB, CD are parallels by Prop. 28. B. 1. in the fame manner AD, BC are parallels. therefore ABCD is a parallelogram, and its oppofite fides are equal by 34. Prop. B. 1. B PROP. VII. B. I. C There are two cafes of this Propofition, one of which is not in the Greek text, but is as necessary 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 Demonstration of this cafe, can be of no ufe at all in the Elements, or any where elfe, but in this Demonftration; 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 stand in need of it except the 7. Propofition, on account of which it has been added. befides the translation from the Arabic has this cafe explicitely demonstrated. and Proclus acknowledges that the second 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 exprefly 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 enuntiation of this 7. Prop. fo as to preferve the very fame meaning; the literal translation from the Greek being extremely harsh, and difficult to be understood by beginners. PROP. PROP. XI. B. I. A Corolloray is added to this Propofition, which is necessary 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 Demonstration; and his anfwer is, that tho' the truth of it be manifest to our fenfes, yet it is science which must give the reason why two fides of a triangle are greater than the third. but the right anfwer to this objection against this and the 21. and fome other plain Propofitions, is, that the number of Axioms ought not to be encreased without neceffity, as it must be if thefe 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 bafe; 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 less than that of two to one. as Pappus Alexandrinus has demonftrated in Prop. 3. B. 3. of his Mathematical Collections. PROP. XXII. B. I. Book I. Some Authors blame Euclid because he does not demonftrate that the two circles made use of in the construction of this Problem must cut one another. but this is very plain from the determination he has given, viz. that any two of the straight lines DF, FG, GH` must be greater than the third. for who is fo dull, tho' only beginning to learn the Elements, as not to perceive that the circle defcribed from the center F, at the distance FD, must meet FH betwixt F and H, because FD is lefs than FH; and that, for the like reafon, the circle defcribed from the center G, at the distance GH or GM must meet DG betwixt D and G; and DM F G T 3 H that Book I. that these circles must meet one another, because FD and GH are DM F G H together greater than FG? and this PROP. XXIV. B. I. D To this is added" of the two fides DE, DF, let DE be that Mr. Thomas Simpfon in p. 262. of the that the point F falls below the line EG; this probably Euclid omitted, as it is very easy to perceive that DG being equal to DF, the point G is in the circumference of a circle defcribed from the center 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. The Propofition which is ufually called the 5. Poftulate, or 11. Axiom, by fome the 12. on which this 29. depends, has given a great great deal to do both to antient and modern Geometers, it seems Book I. not to be properly placed among the Axioms, as, indeed, it is not felf-evident; but it may be demonstrated thus. DEFINITION 1. The distance of a point from a straight line, is the perpendi cular drawn to it from the point. One ftraight line is faid to go nearer to, or further from another ftraight line, when the distance of the points of the first from the other straight line become lefs or greater than they were; and two straight lines are faid to keep the fame distance from one another, when the distance of the points of one of them from the other is always the fame. AXIOM. A ftraight line cannot first come nearer to another straight line, and then go further from it, be A B C E ther from another ftraight line, and F G H then come nearer to it; nor can a straight line keep the fame diftance from another ftraight line, and then come nearer to it, or go further from it; or a straight line keeps always the fame direction. For example, the straight line ABC cannot first come nearer to F G H in like manner, the ftraight line FGH cannot 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 laft cafe as in fig. 2. PROP. I. If two equal ftraight lines AC, BD be each at right angles to the fame straight line AB; if the points C, D be joined by the straight 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 than CA, the |