289 NOTES, &c. DEFINITION I. BOOK I. It is necessary to consider a solid, that is, a magnitude Book I. which has length, breadth, and thickness, in order to understand aright the definitions of a point, line, and superficies; for these all arise from a solid, and exist in it: The boundary, or boundaries which contain a solid are called superficies, or the boundary, which is common to two solids which are contiguous, or which divides one solid into two contiguous parts, is called a superficies : Thus, if BCGF be one of the boundaries which contain the solid ABCDEFGH, or which is the common boundary of this solid, and the solid BKLCFNMG, and is therefore in the one as well as in the other solid, is called a superficies, and has no thickness : For, if it have any, this thickness H G M must either be a part of the thickness of the solid AG, or of the so- € F N lid BM, or a part of the thickness of each of them. It cannot be a part of the thickness of the solid D IC UL BM; because if this solid be removed from the solid AG, the superficies BCGF, the boundary B A of the solid AG, remains still the same as it was. Nor can it be a part of the thickness of the solid AG; because if this be removed from the solid BM, the superficies BCGF, the boundary of the solid BM does nevertheless remain; therefore the superficies BCGF has no thickness, but only length and breadth. The boundary of a superficies is called a line, or a line is the common boundary of two superficies that are contiguous, or which divides one superficies into two contiguous parts: Thus if BC be one of the boundaries which contain the superficies ABCD, or which is the common boundary of this superficies, and of the superficies KBCL which is contiguous to it, this boundary BC is called a line, and has no breadth: For if it have any, this must be part either of the breadth of the superficies ABCD, or of the superficies KBCL, or part of each of them. It is not part of the breadth of the superficies KBCL: for, if this superficies be U А Boox I. removed from the superficies ABCD, the line BC, which is the boundary of the superficies ABCD, remains the same The boundary of a line is called a point, or a point is the H G M B K DEF. VII. B. I. DEF. VIII. B. I. the meaning of the words to subelaç that is, in a straight Book I. line, or in the same direction, be plain, when two straight lines are said to be in a straight line, it does not appear wbat 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 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 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 circumference 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, th 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, А D DB are equal to the two CB, BD, and 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; B. 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. D And if the angle BAD be equal to the opposite angle А. B 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. 1. 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 the proposition itself. Whoever is curious may read what Proclus says of this in his commentary on the 5th and 7th Propositions; for it is not worth while to relate his trifles at full length. It was thought proper to change the enunciation of this 7th Prop. so as to preserve the very same meaning; the literal translation from the Greek being extremely harsh, and difficult to be understood by beginners. |