the meaning of the words sn'sv}six;, that is, in a straight Book I. Iine, or in the same direction, be plain, when two straight lines are said 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 said to be in the same direetion; 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 said of the angle of a semicircle, and the angles of segments, in the 16th and 31st propositions of Book 3, are the additions of some less skilful editor: On which account, especially since they are quite useless, these definitions are distinguished from the rest by inverted double commas. DEF. XVII. B. I. THE words “which also 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 demonstrates 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 centre to the circumferenee would not be all equal: The same thing is easily deduced from the 31st Prop. of Book 3, and the 24th of the same; from the first of which it follows, that semicircles are similar segments of a circle; and from the other, that they are equal to one another. , DEF. XXXIII. B. I. This definition has one condition more than is necessary; because every quadrilateral figure which has its opposite sides equal to one another, has likewise its opposite angles equal; and on the contrary. Let ABCD be a quadrilateral figure, of which the opposite sides AB, CD, are equal to one another; as also AD and BC; join BD; the two sides AD, DB are equal to the two CB, BD, and A. i) the base AB is equal to the base CD; therefore, by Prop. 8. of Book 1. the angle ADB is equal to the angle CBD; P C and, by Prop. 4, B. l. 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. And if the angle BAD be equal to the opposite angle *Y* BCD, and the angle ABC to ADC; the opposite sides are equal: Because, by Prop. 32, B. l. all the angles of the quadrilateral figure ABCD are toge- D 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. l. In the same manner, AD, BC are parallels: Therefore ABCD is a parallelogram, and its opposite sides are equal, by 34th Prop. B. l. PROP. VII. B. I. THERE are two cases of this proposition, one of which is not in the Greek text, but is as necessary as the other: And that the case left out has been formerly in the text, appears plainly from this, that the second part of Prop. 5. which is necessary to the demonstration of this case, can be of no use at all in the Elements, or any where else, but in this demonstration; because the second part of Prop. 5. clearly follows from the first part, and Prop. 13. B. l. This part must therefore have been added to Prop. 5. upon account of some proposition betwixt the 5th and 13th, but none of these stand in need of it except the 7th Proposition, on account of which it has been added: Besides, the translation from the Arabic has this case explicitly demonstrated. And Proclus acknowledges, that the second Part of Prop. 5. was added upon account of Prop. 7, but gives a ridiculous reason for it, “that it might afford an answer to objections “made against the 7th,” as if the case of the 7th, which is left out, were, as he expressly makes it, an objection against PROP. XI. B. I. A sorollary is added to this proposition, which is necessary to Prop. 1. B. XI. and otherwise. PROP. XX. and XXI. B.I. PRoelus, in his commentary, relates, that the Epicureans derided this proposition, as being manifest even to asses, and needing no demonstration; and his answer is, that though the truth of it be manifest to our senses, yet it is science which must give the reason why two sides of a triangle are greater than the third: But the right answer to this objection against this and the 21st, and some other plain propositions, is, that the number of axioms ought not to be increased without necessity, as it must be if these propositions be not demonstrated. Mons. Clairault, in the Preface to his Elements of Geometry, published in French at Paris, anno 1741, says, “That Euclid has been at the pains to prove, that the two sides of a triangle which is included within another, are together less than the two sides of the triangle which includes it:” But he has forgot to add this condition, viz. that the triangles must be upon the same base: because, unless this be added, the sides of the included triangle may be greater than the sides of the triangle which includes it, in any ratio which is less than that of two to one, as Pappus Alexandrinus has demonstrated in Prop. 3. B. 3. of his mathematical collections. PROP, XXII. B. l. SoME authors blame Euclid because he does not demonstrate 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 so dull, though only beginning to learn the Elements, as not to perceive that the l circle joi from the I) M G H ceatre F, at the distance FD, must meet FH betwixt F and H, because FD is less than FH ; and that, for the like reason, the circle described from the centre G, at the distance GH or GM, must meet DG betwixt D and G.; and Book I. Book I. that these circles must meet one another, because FD and STY-' GH are together greater than FG 2 And this determination is easier to be understood than that which Mr. Thomas Simpson derives from it, and uts instead of Euclid's, in }. 49th page of his Ele- D M F. G. ments of Geometry, that he may supply the omission he blames Euclid for, which determination is, that any of the three straight lines must be less than the sum, but greater than the difference of the other two: From this he shows the circles must meet one another, in one case; and says, that it may be proved after the same manner in any other case: But the straight line GM, which he bids take from GF may be greater than it, as in the figure here annexed; in which case his demonstration must be changed into another. PROP. XXIV. B. I. To this is added, “of the two sides DE, DF, let DE be “that which is not greater than the other;” that is, take that side of the two DE, DF which is not greater than the other, in order to make with it the angle EDG equal to BAC; because with- D out this restriction there might be three different cases of the proposition, as Campanus and others make. Mr. Thomas Simpson, in p.262 of the second edition of his Elements of Geometry, printed anno 1760, observes in his notes, that it E. G ought to have been shown, that the oint F falls below the line EG. F 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 described from the centre D at the distance DF, and must be in that part of it which is above the straight line EF, because DG falls above DF, the angle EDG being greater than the angle EDF. PROP. XXIX. B. I. THE proposition which is usually called the 5th postulate, or 11th axiom, by some the 12th, on which this 29th depends, has given a great deal to do, both to ancient and modern geometers: It seems not to be properly placed Pook I. among the axioms, as indeed it is not self-evident; but it may be demonstrated thus: DEFINITION I. The distance of a point from a straight line, is the perpendicular drawn to it from the point. ONE straight line is said to go nearer to, or further from, another straight line, when the distances of the points of the first from the other straight line become less or greater than they were; and two straight lines are said to keep the same distance from one another, when the distance of the points of one of them from the other is always the same. AXIOM. straight line, and then go fur- A C ther from it, before it cuts it; T – B – and, in like manner, a straight D - E. line cannot go further from an- F --> −H other straight line, and then G come nearer to it; nor can a straight line keep the same distance from another straight line, and then come nearer to it, or go further from it; for a straight line keeps always the same direction. For example, the straight line ABC cannot first come nearer to the straight line DE, as - from the point A to the point B, *—”— See the and then, from the point B to the P 7 above point C, go further from the same FTC-H figure. PE: And, in like manner, the straight line FGH cannot go further from DE, as from F to G, and then from G to H, come nearer to the same DE: And so in the last case, as in fig. 2. PROP. H. If two equal straight lines AC, BD, be each at right angles to the same straight line AB: If the points C, D be s joined by the straight line CD, the straight line EF drawn from any point E in AB unto CD, at right angles to AB, shall be equal 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 less than CA, the straight line CFD is nearer to the |