Book IV. PROP. XIV. PRO B. O defcribe a circle about a given equilateral and Let ABCDE be the given equilateral and equiangular pentagon; it is required to defcribe a circle about it. E Bifect a the angles BCD, CDE by the ftraight lines CF, a 9. 1. FD, and from the point F, in which they meet, draw the ftraight lines FB, FA, FE to the points B, A, E. It may be demonftrated, in the fame manner as in the preceding propofition, that the angles CBA, BAE, AED are bifected by the ftraight lines. FB, FA, FE: and because the angle BCD is equal to the angle CDE, and that FCD is the half of the angle BCD, and CDF the half of ČDE; the angle FCD B b is equal to FDC; wherefore the fide CF is equal to the b 6. 1. fide FD: In like manner it may be demonstrated, that FB, FA, FE are each of them equal to FC or FD: therefore the five straight lines FA, FB, FC, FD, FE are equal to one another; and the circle defcribed from the centre F, at the distance of one of them, fhall pass through the extremities of the other four, and be defcribed about the equilateral and equiangular pentagon ABCDE. Which was to be done. PROP. XV. PROB. To infcribe an equilateral and equiangular hex agon in a given circle. Let Book IV. BC in E; therefore BE, EC are, each of them, the fifteenth part of the whole circumference ABCD: therefore if the ftraight lines BE, EC be drawn, and straight lines equal to them be placed around in the whole circle, an equilateral and equiangular quindecagon will be infcribed in it. Which was to be done. d 1. 4. And in the fame manner as was done in the pentagon, if through the points of divifion made by infcribing the quinde cagon, ftraight lines be drawn touching the circle, an equilateral and equiangular quindecagon may be described about it: And likewife, as in the pentagon, a circle may be infcribed in a given equilateral and equiangular quindecagon, and cir cumfcribed about it. ELE ELEMENTS OF GEOMETRY. BOOK V. N the demonstrations of this book there are certain figns Book V. or characters which it has been found convenient to "IN employ. 1. The letters A, B, C, &c. are used to denote magnitudes of any kind. The letters m, n, p, q, are used to denote numbers only. 2. The fign (plus), written between two letters, that denote magnitudes or numbers, fignifies the fum of those magnitudes or numbers. Thus, A+B is the fum of the two magnitudes denoted by the letters A and B; m + n is the fum of the numbers denoted by m and n. 3. The fign-(minus), written between two letters, fignifies the excess of the magnitude denoted by the first of these letters, which is fuppofed the greatest, above that which is denoted by the other. Thus, A-B fignifies the excess of the magnitude A above the magnitude B. 4. When a number, or a letter denoting a number, is written close to another letter denoting a magnitude of any kind, Book V. it fignifies that the magnitude is multiplied by the number. Thus, 3A fignifies three times A; mB, m times B, or a multiple of B by m. When the number is intended to muftiply two or more magnitudes that follow, it is written thus, m.A+B. which fignifies the fum of A and B taken m times; m.A—B is m times the excefs of A above B. Also, when two letters that denote numbers are written close to one another, they denote the product of those numbers, when multiplied into one another. Thus, mn is the product of m into n; and mnA is A multiplied by the product of m into n. 5. The fignfignifies the equality of the magnitudes denoted by the letters that ftand on the oppofite fides of it; A B fignifies that A is equal to B: A+B=C-D fignifies that the fum of A and B is equal to the excess of C above D. 6. The fignis used to fignify the inequality of the magnitudes between which it is placed, and that the magnitude to which the opening of the lines is turned is greater than the other. Thus A B fignifies that A is greater than B; and AB fignifies that A is lefs than B." A DEFINITIONS. I. LESS magnitude is faid to be a part of a greater magnitude, when the less measures the greater, that is, when the lefs is contained a certain number of times, exactly, in the greater. II. A greater magnitude is faid to be a multiple of a lefs, when the greater is measured by the lefs, that is, when the greater contains the lefs a certain number of times exactly. III. Ratio is a mutual relation of two magnitudes, of the fame kind, to one another, in refpect of quantity. IV. Magnitudes are said to be of the fame kind, when the less can be multiplied fo as to exceed the greater; and it is only fuch magnitudes that are faid to have a ratio to one another. V. v. Book V. If there be four magnitudes, and if any equimultiples what- See N. foever be taken of the firft and third, and any equimultiples whatsoever of the fecond and fourth, and if, according as the multiple of the firft is greater than the multiple of the fecond, equal to it, or lefs, the multiple of the third is also greater than the multiple of the fourth, equal to it, or lefs; then the first of the magnitudes is faid to have to the fecond the fame ratio that the third has to the fourth. VI. Magnitudes are faid to be proportionals, when the first has the VII. When of the equimultiples of four magnitudes (taken as in the fifth definition) the multiple of the firft is greater than that of the fecond, but the multiple of the third is not greater than the multiple of the fourth; then the firft is faid to have to the fecond a greater ratio than the third magnitude has to the fourth; and, on the contrary, the third is faid to have to the fourth a lefs ratio than the firft has to the fecond. VIII. When there is any number of magnitudes greater than two, of which the firft has to the fecond the fame ratio that the fecond has to the third, and the fecond to the third the fame ratio which the third has to the fourth, and fo on, the magnitudes are faid to be continual proportionals. IX. When three magnitudes are continual proportionals, the fecond is faid to be a mean proportional between the other two. |