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T is necessary to consider a solid, that is a magnitude 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 folid into two contiguous parts, is called a superficies. thus if BCGF be one of the boundaries which contain the folid ABCDEFGH, or which is the common boundary of this solid, and the solid BKLCFNMG and is therefore in the one as well as the other solid, it is called a superficies, and has no thickness. for if it have any, this thickness must either be a part of the 'thickness of the solid AG, or of the
H G M folid BM, or a part of the thickness
F of each of them. It cannot be
part of the thickness of the folid BM, because if this folid be removed from
L the solid AG, the superficies BCGF, the boundary of the solid AG, remains fill the fame as it was. Nor A В к. can it be a part of the thickness of the folid 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 removed from the superficies ABCD, the line BC which is the boundary of the superficies ABCD
Book I. remains the same as it was. nor can the breadth that BC is sup
posed to have be a part of the breadth of the superficies ABCD, because if this be removed from the superficies KBCL, the line BC which is the boundary of the superficies KBCL does nevertheless remain. therefore the line BC has no breadth, and because the line BC is in a superficies, and that a superficies has no thickness, as was fhewn; therefore a line has neither breadth ner thickness, but only length.
The boundary of a line is called a point, or a point is the com-
line A B K
DEF. VII. B. I. Instead of this Definition as it is in the Greek copies, a more distinct one is given from a property of a plane superficies, which is manifestly supposed in the Elements, viz. that a straight line drawn from any point in a plane to any other in it, is wholly in that plane,
DEF. VIII. B. I.
It seems that he who made this Definition designed that it should comprehend not only a plane angle contained by two straight lines, but likewise the angle which some conceive to be made by a straight line and a curve, or by two curve lines, which meet one another in a plane. but tho' the meaning of the words éx' gideias; that is,
in a straight line, or in the same direction, be plain, when two Book I. 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 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 16. and 31. 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 or 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 Co. rollary 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 center to the circumference would not be all equal. the fame thing is easily deduced from the 31. Prop. of Book the 24. of the fame ; from the first of which it follows that semicircles are similar segments of a circles and from the other, that they are equal to one another.
DÉF. XXXIII. B. I. This Definition has one condition more than is necessary; be. cause 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 fides AB, CD are equal to one another; as also AD and BC. join BD; the two fides AD, DB are equal to the two CB, BD, and the base AB is equal to the base CD; therefore by Prop. 8. of B 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.
And if the angle BAD be equal to the opposite angle BCD, and the angle ABC to ADC; the opposite sides 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 to
A gether equal to the two angles BCD, ABC. wherefore BAD, ADC are the half of all the four angles; that is, BAD B and ADC are equal to two right angles. and therefore AB, CD are parallels by Prop. 28. B. 1. in the same manner AD, BC are parallels. therefore ABCD is a parallelogram, and its opposite fides are equal by 34. Prop. B. 1.
PROP. VII. B. I.
There are two cases of this Proposition, 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 second part of Prop 5. which is necessary to the Demonftration 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.-I. this part
must therefore have been added to Prop. 5. upon account of some Proposition betwixt the 5. and 13. but none of these stand in need of it, except the 7. Proposition, on account of which it has been added. besides the translation from the Arabic has this case explicitely 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 7." as if the case of the 7. 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 5. and 7. Propositions; 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. so as to preserve the very fame meaning; the literal translation from the Greek being extremely harsh, and difficult to be understood by beginners.