a straight line be drawn at right angles to the touching line, the centre of the circle is in that line. XX. The angle at the centre of a circle is double the angle at the circumference, upon the same base, that is, upon the same point of the circumference. XXI. The angles in the same segment of a circle are equal to one another. XXII. The opposite angles of any quadrilateral figure described in a circle, are together equal to two right angles. XXIII. Upon the same straight line, and upon the same side of it, there cannot be two similar segments of circles, not coinciding with one another. XXIV. Similar segments of circles upon equal straight lines are equal to one another. XXVI. In equal circles, equal angles stand upon equal arches, whether they be at the centres or circumferences. XXVII. In equal circles, the angles which stand upon equal arches are equal to one another, whether they be at the centres or circumferences. XXVIII. In equal circles, equal straight lines cut off equal arches, the greater equal to the greater, and the less to the less. XXIX. In equal circles equal arches are subtended by equal straight lines. XXXI. In a circle, the angle in a simicircle is a right angle; but the angle in a segment greater than a simicircle is less than a right angle; and the angle in a segment less than a simicircle is greater than a right angle. Cor. From this it is manifest, that if one angle of a triangle be equal to the other two, it is a right angle, because the angle adjacent to it is equal to the same two; and when the adjacent angles are equal, they are right angles. XXXI. If a straight line touch a circle, and from the point of contact a straight line be drawn cutting the circle, the angles made by this line with the line which touches the circle, shall be equal to the angles in the alternate segments of the circle. XXXIV. If two straight lines within a circle cut one another, the rectangle contained by the segments of one of them is equal to the rectangle contained by the segments of the other. XXXVI. If from any point without a circle two straight lines be drawn, one of which cuts the circle, and the other touches it; the rectangle contained by the whole line which cuts the circle, and the part of it without the circle, is equal to the square of the line which touches it. XXXVII. If from a point without a circle there be drawn two straight lines, one of which cuts the circle, and the other meets it; if the rectangle

contained by the whole line, which cuts the circle, and the part of it without a circle, be equal to the line which meets it, the line which meets shall touch the circle.

BOOK IV.--DEFINITIONS. 1. A rectilineal figure is said to be inscribed in another rectilineal figure, when all the angles of the inscribed figure are upon the sides of the figure in which it is inscribed, each upon each. 2. In like manner, a figure is said to be described about another figure, when all the sides of the circumscribed figure pass through the angular points of the figure about which it is described, each through each. 3. A rectilineal figure is said to be inscribed in a circle, when all the angles of the inscribed figure are upon the circumference of the circle. 4. A rectilineal figure is said to be described about a circle, when each side of the circumscribed figure touches the circumference of the circle. 5. In like manner, a circle is said to be inscribed in a rectilineal figure, when the circumference of the circle touches each side of the figure. 6. A circle is said to be described about a rectilineal figure, when the circumference of the circle passes through all the angu. lar points of the figure about which it is described. 7. A straight line is said to be placed in a circle, when the extremities of it are in the circumference of the circle.

BOOK V.--DEFINITIONS. 1. 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. 2. 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. 3. Ratio is a mutual relation of two magnitudes, of the same kind, to one another, in respect of quantity. 4. 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 the same ratio to one another. 5. 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. 6. Magni. tudes 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. 7. 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 then the first has to the second. 8. When there is any number of magnitudes greater than two, of which the first has to the second the same ratio which 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. 9. When three magnitudes are continual proportionals, the second is said to be a mean proportional between the other two. 10. When there are 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 to the last magnitude. 11. 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. 12. 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. 13. In proportionals, the antecedent terms are called homologous to one another, as also the consequents 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 may continue still to be proportionals. 15. Permutando, or alternando, by permuta.

tion, 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. 16. 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. 16. 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. 17. Dividendo, or division: when there are four proportionals, and it is inferred, that the excess of the first above the second, is to the second, as the excess of the third above the fourth, is to the fourth. 18. 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 is to its excess above the fourth. 19. 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 dif ferent order in which the magnitudes are taken two and two. 20. 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. 21. Ex æquali, in proportione in 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 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 in a cross, or inverse, order; and the inference is as in the 19th definition.

AXIOMS. 1. Equimultiples of the same or of equal magnitudes, are equal to one another. 2. Those magnitudes of which the same or equal magnitudes, are equimultiples, are equal to one another. 3. A multiple of a greater magnitude is greater than the same multiple of a less. 4. That magnitude of which a multiple is greater than the same multiple of another, is greater than that other magnitude.

PROP. I. If any number of magnitudes be equimultiples of as many others, each of each, what multiple soever any one of the first is of its part, the same multiple is the sum of all the first of the sum of all the rest. II. If to a multiple of a magnitude by any number, a multiple of the same magnitude by any number be added, the sum will be the same multiple of that magnitude that the sum of the two numbers is of unity. III. If the first of three magnitudes contain the second as oft as there are units in a certain number, and if the second contain the third also, as often as there are units in a certian number, the first will contain the third as oft as there are units in the product of these two numbers. IV. If the first of four magnitudes has the same ratio to the second which the third has to the fourth, and if any equimultiples whatever be taken of the first and third, and any whatever of the second and fourth; the multiple of the first shall have the same ratio to the multiple of the second, that the multiple of the third has to the multiple of the fourth. V. If one magnitude be the same multiple of another, which a magnitude taken from the first is of a magnitude taken from the other; the remainder is the same multiple of the remainder, that the whole is of the whole. VI. If from a multiple of a magnitude by any number a multiple of the same magnitude by a less number be taken away, the remainder will be the same multiple of that magnitude that the difference of the numbers is of unity. A. If four magnitudes be proportionals, they are proportionals also when taken inversely. B. If the first be the same multiple of the second, or the same part of it, that the third is of the fourth; the first is to the second as the third to the fourth. C. If the first be to the second as the third to the fourth; and if the first be a multiple or a part of the second, the third is the same multiple or the same part of the fourth. VII. Equal magnitudes have the same ratio to the same magnitude; and the same has the same ratio to equal magnitudes. VIII. Of