equal to the straight line from the centre, that is, to the radius of the circle.

And if, through the points A, B, C, D, E, F, there be drawn straight lines touching the circle, an equilateral and equiangular hexagon shall be described about it, which may be demonstrated from what has been said of the pentagon; and likewise a circle may be inscribed in a given equilateral and equiangular hexagon, and circumscribed about it, by a method like to that used for the pentagon.

PROP. XVI. PROB.

To inscribe an equilateral and equiangular quindecagon in a given circle.

Let ABCD be the given circle; it is required to inscribe an equilateral and equiangular quindecagon in the circle ABCD

Let AC be the side of an equilateral triangle inscribed (IV. 2) in the circle, and AB the side of an equilateral and equiangular pentagon inscribed (IV. 11) in the same; therefore, of such equal B parts as the whole circumference ABCDF contains fifteen, the arc ABC, being the F third part of the whole, contains five; and the arc AB, which is the fifth part of the whole, contains three; therefore BC, their difference, contains two of the same

parts: bisect (III. 30) BC in E; therefore BE, EC are each of them the fifteenth part of the whole circumference ABCD: therefore if the straight lines BE, EC be drawn, and straight lines equal to them be placed (IV. 1) around in the whole circle, an equilateral and equiangular quindecagon will be inscribed in it. Which was to be done.

And, in the same manner, as was done in the pentagon, if through the points of division made by inscribing the quindecagon, straight lines be drawn touching the circle, an equilateral and equiangular quindecagon may be described about it; and likewise, as in the pentagon, a circle may be inscribed in a given equilateral and equiangular quindecagon, and circumscribed about it.

BOOK FIFTH.

In the demonstrations of this book there are certain signs or characters which it has been found convenient to employ.

I. 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. II. The sign + (plus), written between two letters that denote magnitudes or numbers, signifies the sum of those magnitudes or numbers. Thus A+B is the sum of the two magnitudes denoted by the letters A and B; m+n is the sum of the numbers denoted by m and n.

III. The sign (minus), written between two letters, signifies the excess of the magnitude denoted by the first of these letters, which is supposed the greatest, above that which is denoted by the other. Thus A-B signifies the excess of the magnitude A above the magnitude B.

IV. When a number, or a letter denoting a number, is written close to another letter denoting a magnitude of any kind, it signifies that the magnitude is multiplied by the number. Thus, 3A signifies three times A; mB, m times B, or a multiple of B by m. When the number is intended to multiply two or more magnitudes that follow, it is written thus, m(A+B), which signifies the sum of A and B taken m times; m(A-B) is m times the excess 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.

V. The sign = signifies the equality of the magnitudes denoted by the letters that stand on the opposite sides of it; A=B signifies that A is equal to B; A+B=C-D signifies that the sum of A and B is equal to the excess of C above D.

VI. The sign is used to signify that the magnitudes between which it is placed are unequal, and that the magnitude to which the opening of the lines is turned is greater than the other. Thus A B signifies that A is greater than B; and A≤B signifies that A is less than B.

DEFINITIONS.

I. A less magnitude is said to be a part of a greater magnitude, when the less measures the greater; that is, when the less is contained a certain number of times exactly in the greater.

II. A greater magnitude is said to be a multiple of a less, when the greater is measured by the less, that is, when the greater contains the less a certain number of times exactly.

III. Ratio is a mutual relation of two magnitudes, of the same kind, to one another in respect of quantity.

IV. Magnitudes are said to be of the same kind, when the less can be multiplied so as to exceed the greater; and it is only such magnitudes that are said to have a ratio to one another. V. If there be four magnitudes, and if any equimultiples whatsoever be taken of the first and third, and any equimultiples whatsoever of the second and fourth, and if, according as the multiple of the first is greater than the multiple of the second, equal to it, or less, the multiple of the third is also greater than the multiple of the fourth, equal to it, or less; then the first of the magnitudes is said to have to the second the same ratio that the third has to the fourth.*

VI. Magnitudes are said to be proportionals, when the first has the same ratio to the second that the third has to the fourth; and the third to the fourth the same ratio which the fifth has to the sixth, and so on, whatever be their number. When four magnitudes, A, B, C, D are proportionals, it is usual to say that A is to B as C to D, and to write them thus, A:B::C:D; or thus, A: B=C : D.

VII. When of the equimultiples of four magnitudes, taken as in the fifth definition, the multiple of the first is greater than that of the second, but the multiple of the third is not greater than the multiple of the fourth; then the first is said to have to the second a greater ratio than the third magnitude has to the fourth; and, on the contrary, the third is said to have to the fourth a less ratio than the first has to the second.

VIII. When there is any number of magnitudes greater than two, of which the first has to the second the same ratio that the second has to the third, and the second to the third the same ratio which the third has to the fourth, and so on, the magnitudes are said to be continual proportionals.

IX. When three magnitudes are continual proportionals, the second is said to be a mean proportional between the other two. X. When there is any number of magnitudes of the same kind, the first is said to have to the last the ratio compounded of the ratio which the first has to the second, and of the ratio which the second has to the third, and of the ratio which the third has to the fourth, and so on unto the last magnitude. For example, if A, B, C, D be four magnitudes of the same kind, the first A is said to have to the last D, the ratio compounded of the ratio of A to B, and of the ratio of B to C, and of the

*See Notes.

ratio of C to D; or the ratio of A to D is said to be compounded of the ratios of A to B, B to C, and C to D.

And if A: B:: E: F; and B:C:: G: H; and C: D::K: L, then, since by this definition, A has to D the ratio compounded of the ratios of A to B, B to C, C to D: A may also be said to have to D the ratio compounded of the ratios which are the same with the ratio of E to F, G to H, and K to L.

In like manner, the same things being supposed, if M has to N the same ratio which A has to D, then, for shortness' sake, M is said to have to N a ratio compounded of the same ratios, which compound the ratio of A to D; that is, a ratio compounded of the ratios of E to F, G to H, and K to L.

XI. If three magnitudes are continual proportionals, the ratio of the first to the third is said to be duplicate of the ratio of the first to the second.

Thus, if A be to B as B to C, the ratio of A to C is said to be duplicate of the ratio of A to B. Hence, since by the last definition, the ratio of A to C is compounded of the ratios of A to B, and B to C, a ratio, which is compounded of two equal ratios, is duplicate of either of these ratios.

XII. If four magnitudes are continual proportionals, the ratio of the first to the fourth is said to be triplicate of the ratio of the first to the second, or of the ratio of the second to the third, &c. So also, if there are five continual proportionals, the ratio of the first to the fifth is called quadruplicate of the ratio of the first to the second; and so on, according to the number of ratios. Hence, a ratio compounded of three equal ratios is triplicate of any one of those ratios; a ratio compounded of four equal ratios, quadruplicate, &c.

XIII. In proportionals, the antecedent terms of the ratios are said to be homologous to one another, and the consequents of the ratios are said to be homologous to one another.

Geometers make use of the following technical words to signify certain ways of changing either the order or magnitude of proportionals, so as that they continue still to be proportionals :XIV. Permutando, or alternando, by permutation, or alternately: This word is used when there are four proportionals, and it is inferred, that the first has the same ratio to the third which the second has to the fourth; or that the first is to the third as the second to the fourth. See Prop. 16 of this Book.

XV. Invertendo, by inversion: When there are four proportionals, and it is inferred, that the second is to the first as the fourth to the third. Prop. A., Book 5.

XVI. Componendo, by composition: When there are four proportionals, and it is inferred, that the first, together with the second, is to the second as the third, together with the fourth, is to the fourth. 18th Prop., Book 5.

XVII. Dividendo, by division: When there are four proportionals, and it is inferred, that the excess of the first above the second to the second, as the excess of the third above the fourth is to the fourth. 17th Prop., Book 5.

XVIII. Convertendo, by conversion: When there are four proportionals, and it is inferred, that the first is to its excess above the second as the third to its excess above the fourth. Prop. D., Book 5.

XIX. Ex æquali (sc. distantia), or ex æquo, from equality of distance: When there is any number of magnitudes more than two, and as many others, so that they are proportionals when taken two and two of each rank, and it is inferred, that the first is to the last of the first rank of magnitudes as the first is to the last of the others. Of this there are the two following kinds, which arise from the different order in which the magnitudes are taken two and two.

XX. Ex æquali, from equality: This term is used simply by itself, when the first magnitude is to the second of the first rank as the first to the second of the other rank; and as the second is to the third of the first rank, so is the second to the third of the other; and so on in order, and the inference is as mentioned in the preceding definition; whence this is called ordinate proportion. It is demonstrated in the 22d Prop., Book 5.

XXI. Ex æquali, in proportione perturbata seu inordinata, from equality, in perturbate or disorderly proportion: This term is used when the first magnitude is to the second of the first rank as the last but one is to the last of the second rank; and as the second is to the third of the first rank, so is the last but two to the last but one of the second rank; and as the third is to the fourth of the first rank, so is the third from the last to the last but two of the second rank; and so on in a cross, or inverse order; and the inference is as in the 19th definition. It is demonstrated in the 23d Prop. of Book V.

AXIOMS.

I. Equimultiples of the same, or of equal magnitudes, are equal to one another.

II. Those magnitudes of which the same, or equal magnitudes, are equimultiples, are equal to one another.

III. A multiple of a greater magnitude is greater than the same multiple of a less.

IV. The magnitude of which a multiple is greater than the same multiple of another, is greater than that other magnitude.

F