if M has to N the same ratio which A has to D; then, for shortness sake, M is said to have to N, the ratio compounded of the ratios of E to F, G to H, and K to L

12. 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 continue still to be proportionals.

13. 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: as is shewn in the XVI. Prop. of this Book V.

14. 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. B. Book V.

15. 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. Prop. XVIII. Book V.

16. Dividendo, by 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. Prop. XVII. Book V.

17. Convertendo, by conversion; when there are four proportionals, and it is inferred, that the first is to its excess above the secoud, as the third to its excess above the fourth. Prop. E. Book V.

18. Ex aequali (sc. distantia), or ex aequo, 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.

19. Ex aequali, 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 Prop. XXII. Book V.

20. Ex aequali, 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 order: and the inference is as in Def. 18. It is demonstrated in Prop. XXIII. Book V.

4. That magnitude of which a multiple is greater than the same multiple of another, is greater than that other magnitude.

Prop. I. Theor. If any number of magnitudes be equinultiples of as many, each of each; what multiple soever any one of them is of its part, the same multiple shall all the first magnitudes be of all the other.

Prop. II. Theor. If the first magnitude be the same multiple of the second that the third is of the fourth, and the fifth the same multiple of the second that the sixth is of the fourth; then shall the first together with the fifth be the same multiple of the second, that the third together with the sixth is of the fourth.

Prop. II. Theor. If the first be the same multiple of the second, which the third is of the fourth; and if of the first and third there be taken equimultiples, these shall be equimultiples, the one of the second, and the other of the fourth.

Prop. IV. Theor. If the first of four magnitudes has the same ratio to the second which the third has to the fourth; then any equimultiples whatever of the first and third shall have the same ratio to any equimultiples of the second and fourth, viz. the equimultiple of the first shall have the same ratio to that of the second, which the equimultiple of the third has to that of the fourth.

Prop. V. Theor. 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 shall be the same multiple of the remainder, that the whole is of the whole.

Prop. VI. Theor. If two magnitudes be equimultiples of two others, and if equimultiples of these be taken from the first two, the remainders are either equal to these others, or equimultiples

of them.

Prop. A. Theor. If the first of four maguitudes has to the second, the same ratio which the third has to the fourth; then, if the first be greater than the second, the third is also greater than the fourth; and, if equal, equal; if less, less.

Prop. B. Theor. If four magnitudes are proportionals, they are proportionals also when taken inversely.

Prop. C. Theor. 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 is to the fourth.

Prop. D. Theor. If the fist be to the second as the third to the fourth, and if the first be a multi ple, or part of the second; the third is the same inultiple, or the same part of the fourth.

Prop. VII. Theor. Equal magnitudes have the same ratio to the same magnitude; and the same has the same ratio to equal magnitudes.

Prop. VIII. Theor. Of unequal magnitudes, the greater has a greater ratio to the saine than the less has; and the same magnitude has a greater ratio to the less, than it has to the greater.

Prop. IX. Theor. Magnitudes which have the same ratio to the same magnitude are equal to one another; and those to which the same magnitude has the same ratio are equal to one an

Prop. X. Theor. That magnitude which has a greater ratio than another has unto the same magnitude is the greater of the two: and that maguitude to which the same has a greater ratio than it has unto another magnitude is the lesser of the two.

Prop. XI. Theor. Ratios that are the same to the same ratio, are the same to one another.

Prop. XII. Theor. If any number of magnitudes be proportionals, as one of the antecedents is to its consequent, so shall all the antecedents taken together be to all the consequents.

Prop. XIII. Theor. If the first has to the second the same ratio which the third has to the fourth, but the third to the fourth a greater ratio than the fifth has to the sixth; the first shall also have to the second a greater ratie than the fifth has to the sixth.

Prop. XIV. Theor. If the first has to the second, the same ratio which the third has to the fourth; then, if the first be greater than the third, the second shall be greater than the fourth; and if equal, equal; and if less, less.

Prop. XV. Theor. Magnitudes have the same ratio to one another which their equimultiples have.

Prop. XVI. Theor. If four magnitudes of the same kind be proportionals, they shall also be proportionals when taken alternately.

Prop. XVII. Theor. If magnitudes, taken jointly, be proportionals, they shall also be proportion als when taken separately; that is, if two magnitades together have to one of them the same ratio which two others have to one of these, the remaining one of the first two shall have to the other the same ratio which the remaining one of the last two bas to the other of these.

Prob. XVIII. Theor. If magnitudes, taken separately, be proportionals, they shall also be proportionals, when taken jointly, that is, if the first be to the second, as the third to the fourth, the first and second together shall be to the second, as the third and fourth together to the fourth.

Prop. XIX. Theor. If a whole magnitude be to a whole, as a magnitude taken from the first, is to a magnitude taken from the other; the remainder shall be to the remainder, as the whole to the whole.

Prop. E. Theor. If four magnitudes be proportionals, they are also proportionals by conversion, that is, the first is to its excess above the second, as the third to its excess above the fourth.

Prop. XX. Theor. If there be three magnitudes, and other three, which, taken two and two, have the same ratio; if the first be greater than the third, the fourth shall be greater than the sixth; and if equal, equal; and if less, less.

Prop. XXI. Theor. If there be three magnitudes, and other three, which have the same ratio taken two and two, but in a cross order; if the first magnitude be greater than the third, the fourth shall be greater than the sixth; and if equal, equal; and if less, less.

Prop. XXII. Theor. If there be any number of magnitudes, and as many others, which, taken two and two in order, have the same ratio; the first shall have to the last of the first magnitudes the same ratio which the first of the others has to the last. N. B. This is usually cited by the words "ex aequali," or 66 ex aequo."

Prop. XXIII. Theor. If there be any number of magnitudes, and as many others, which, taken two and two, in a cross order, have the same ratio; the first shall have to the last of the first magnitudes the same ratio which the first of the others has to the last. N. B. This is usually cited by the words, "ex aequali in proportione perturbata;” or “ ex aequo perturbate.”

Prop. XXIV. Theor. If the first has to the second the same ratio which the third has to the

fourth; and the fifth to the second, the same ratio which the sixth has to the fourth; the first and fifth together shall have to the second, the same ratio which the third and sixth together have to the fourth.

Prop. XXV. Theor. If four magnitudes of the same kind are proportionals, the greatest and least of them together are greater than the other two together.

Prop. F. Theor. Ratios which are compounded of the same ratios, are the same with one another. Prop. G. Theor. If several ratios be the same with several ratios, each to each; the ratio which is compounded of ratios which are the same with the first ratios, each to each, is the same with the ratio compounded of ratios which are the same with the other ratios, each to each.

Prop. H. Theor. If a ratio compounded of secveral ratios be the same with a ratio compounded of any other ratios, and if one of the first ratios, or a ratio compounded of any of the first, be the same with one of the last ratios, or with the ratio compounded of any of the last; then the ratio compounded of the remaining ratios of the first, or the remaining ratio of the first, if but one remain, is the same with the ratio compounded of those remaining of the last, or with the remaining ratio of the last.

Prop. K. Theor. If there be any number of ratios, and any number of other ratios such, that the ratio compounded of ratios which are the same with the first ratios, each to each, is the same with the ratio eompounded of ratios which are the same each to each, with the last ratios; and if one of the first ratios, or the ratio which is compounded of ratios which are the same with several of the first ratios, each to each, be the same with one of the last ratios, or with the ratio compounded of ratios which are the same, each to each, with several of the last ratios: then the ratio compounded of ratios which are the same with the remaining ratios of the first, each to each, or the remaining ratio of the first, if but one remain; is the same with the ratio compounded of ratios which are the same with those remaining of the last, each to each, or with the remaining ratio of the last.

Book VI. Def. 1.—Similar rectilineal figures are those which have their several angles equal, each to each, and the sides about the equal angles proportionals.

2. Reciprocal figures, viz. triangles and parallelograms, are such as have their sides about two of their angles proportionals in such manner, that a side of the first figure is to a side of the other, as the remaining side of this other is to the remaining side of the first.

3. A straight line is said to be cut in extreme and mean ratio, when the whole is to the greater segment, as the greater segment is to the less. 4. The altitude of any figure is the straight line drawn from its vertex perpendicular to the base.

Prop. I. Theor. Triangles and parallelograms of the same altitude are one to another as their bases.

Prop. II. Theor. If a straight line be drawn parallel to one of the sides of a triangle, it shall cut the other sides, or those produced, proportionally: and if the sides, or the sides produced, be cut proportionally, the straight line which joins the points of section shall be parallel to the remaining side of the triangle.

Prop. III. Theor. If the angle of a triangle be

and similarly described figures upon the sides containing the right angle.

Prop. XXXII. Theor. If two triangles which have two sides of the one proportional to two sides of the other, be joined at one angle, so as to have their homologous sides parallel to one another; the remaining sides shall be in a straight

line.

Prop. XXXIII. Theor. In equal circles, angles, whether at the centres or circumferences, have the same ratio which the circumferences on which they stand have to one another: so also have the sectors.

Prop. B. Theor. If an angle of a triangle be bisected by a straight line, which likewise cuts the base; the rectangle contained by the sides of the triangle is equal to the rectangle contained by the segments of the base, together with the square of the straight line bisecting the angle.

Prop. C. Theor. If from any angle of a triangle a straight line be drawn perpendicular to the base; the rectangle contained by the sides of the trian. gle is equal to the rectangle contained by the perpendicular and the diameter of the circle described about the triangle.

Prop. D. Theor. The rectangle contained by the diagonals of a quadrilateral inscribed in a circle, is equal to both the rectangles contained by its opposite sides.

Book XI. Def. 1.-A solid is that which hath length, breadth, and thickness.

2. That which bounds a solid is a superficies. 3. A straight line is perpendicular, or at right angles to a plane, when it makes right angles with every straight line meeting it in that plane.

4. A plane is perpendicular to a plane, when the straight lines drawn in one of the planes perpendicularly to the common section of the two planes, are perpendicular to the other plane.

5. The inclination of a straight line to a plane is the acute angle contained by that straight line, and another drawn from the point in which the first line meets the plane, to the point in which a perpendicular to the plane drawn from any point of the first line above the plane, meets the same plane.

6. The inclination of a plane to a plane is the acute angle contained by two straight lines drawn from any the same point of their common section at right angles to it, one upon one plane, and the other upon the other plane.

7. Two planes are said to have the same, or a like inclination to one another, which two other planes have, when the said angles of inclination are equal to one another.

8. Parallel planes are such which do not meet one another though produced.

9. A solid angle is that which is made by the meeting of more than two planes, which are not in the same plane, in one point.

10. The tenth definition is omitted for reasons given in the notes.

11. Similar solid figures are such as have all their solid angles equal, each to each, and which are contained by the same number of similar planes.

12. A pyramid is a solid figure contained by planes that are constituted betwixt one plane and one point above it in which they meet.

13. A prism is a solid figure contained by plane figures of which two that are opposite are equal, similar, and parallel to one another; and the others parallelograms.

14. A sphere is a solid figure described by the

cone.

19. The axis of a cone is the fixed straight line about which the triangle revolves.

20. The base of a cone is the circle described by that side containing the right angle, which revolves.

21. A cylinder is a solid figure described by the revolution of a right angled parallelogram about one of its sides which remains fixed.

22. The axis of a cylinder is the fixed straight line about which the parallelogram revolves.

23. The bases of a cylinder are the circles described by the two revolving opposite sides of the parallelogram.

24. Similar cones and cylinders are those which have their axes and the diameters of their bases proportionals,

25. A cube is a solid figure contained by six equal squares.

26. A tetrahedron is a solid figure contained by four equal and equilateral triangles.

27. An octahedron is a solid figure contained by eight equal and equilateral triangles.

28. A dodecahedron is a solid figure contained by twelve equal pentagons which are equilateral and equiangular,

29. An icosahedron is a solid figure contained by twenty equal and equilateral triangles,

Def. A. parallelepiped is a solid figure contained by six quadrilateral figures, whereof every opposite two are parallel.

Prop. I. Theor. One part of a straight line cannot be in a plane and another part above it.

Prop. II, Theor. Two straight lines which cut one another are in one plane, and three straight lines which meet one another are in one plane.

Prop. III. Theor. If two planes cut one another, their common section is a straight line.

Prop. IV. Theor. If a straight line stand at right angles to each of two straight lines in the point of their intersection, it shall also be at right angles to the plane which passes through them, that is, to the plane in which they are.

Prop. V. Theor. If three straight lines meet all in one point, and a straight line stands at right angles to each of them in that point; these three straight lines are in one and the same plane.

Prop. VI. Theor. If two straight lines be at right angles to the same plane, they shall be parallel to one another.

Prop. VII. Theor. If two straight lines be parallel, the straight line drawn from any point in the one to any point in the other, is in the same plane with the parallels.

Prop. VIII. Theor. If two straight lines be parallel, and one of them is at right angles to a plane

the other also shall be at right angles to the same plane.

Prop. IX. Theor. Two straight lines which are each of them parallel to the same straight line, and not in the same plane with it, are parallel to one another.

Prop. X. Theor. If two straight lines meeting one another be parallel to two others that meet one another, and are not in the same plane with the first two; the ûrst two and the other two shall contain equal angles.

Prop. XI. Prob. To draw a straight line perpendicular to a plane, from a given point above it.

Prop. XII. Prob. To erect a straight line at right angles to a given plane, from a point given in the plane.

Prop. XIII. Theor. From the same point in a given plane, there cannot be two straight lines at right angles to the plane, upon the same side of it: and there can be but one perpendicular to a plane from a point above the plane.

Prop. XIV. Theor. Planes to which the same straight line is perpendicular, are parallel to one another.

Prop. XV. Theor. If two straight lines meeting one another, be parallel to two straight lines which meet one another, but are not in the same plane with the first two; the plane which passes through these is parallel to the plane passing through the others.

Prop. XVI. Theor. If two parallel planes be cut by another plane, their common sectious with it are parallels.

Prop. XVII. Theor. If two straight lines be cut by parallel planes, they shall be cut in the same ratio.

Prop. XVIII. Theor. If a straight line be at right angles to a plane, every plane which passes through it shall be at right angles to that plane.

Prop. XIX. Theor. If two planes cutting one another be each of them perpendicular to a third plane; their common section shall be perpendicuJar to the same plane.

Prop. XX. Theor. If a solid angle be contained by three plane angles, any two of them are greater than the third.

Prop. XXI. Theor. Every solid angle is contained by plain angles which together are less than four right angles.

Prop. XXII. Theor. If every two of three plain angles be greater than the third, and if the straight lines which contain them be all equal; a triangle may be made of the straight lines that join the extremities of those equal straight lines.

Prop. XXIII. Prob. To make a solid angle which shall be contained by three given plane angles, any two of them being greater than the third, and all three together less than four right angles.

Prop. A. Theor. If each of two solid angles be contained by three plane angles equal to one another, each to each; the planes in which the equal angles are, have the same inclination to one another.

Prop. B. Theor. If two solid angles be contained, each by three plane angles which are equal to one another, each to each, and alike situated; these solid angles are equal to one another.

Prop. C. Theor. Solid figures contained by the same number of equal and similar planes alike situated, and having none of their solid augies contained by more than three plane angles, are equal and similar to one another.

Prop. XXIV. Theor. If a solid be contained by six planes, two and two of which are parallel; the

opposite planes are similar and equal parallele

grams.

Prop. XXV. Theor. If a solid parallelepiped be cut by a plane parallel to two of its opposite planes; it divides the whole into two solids, the base of one of which shall be to the base of the other, as the one solid is to the other.

Prop. XXVI. Prob. At a given point in a given straight line, to make a solid angle equal to a given solid angle contained by three plane augles.

Prop. XXVII. Prob. To describe from a given straight line a solid parallelepiped similar, and similarly situated to one given.

Prop. XXVIII. Theor. If a solid parallelepiped be cut by a plane passing through the diagonals of two of the opposite planes; it shall be cut in two equal parts.

Prop. XXIX. Theor. Solid parallelepipeds upon the same base, and of the same altitude, the insisting straight lines of which are terminated in the same straight lines in the plane opposite to the base, are equal to one another.

Prop. XXX. Theor. Solid parallelepipeds upon the same base, and of the same altitude, the insisting straight lines of which are not terminated in the same straight lines in the plane opposite to the base, are equal to one another.

Prop. XXXI. Theor. Solid parallelepipeds which are upon equal bases, and of the same altitude, are equal to one another.

Prop. XXXII.Theor. Solid parallelepipeds which have the same altitude, are to one another as their bases.

Prop. XXXIII. Theor. Similar solid parallelepipeds are one to another in the triplicate ratio of their homologous sides.

Prop. D. Theor. Solid parallelepipeds contained by parallelograms equiangular to one auother, each to each, that is, of which the solid angles are equal, each to each, have to one another the ratio which is the same with the ratio compounded of the ratios of their sides.

Prop. XXXIV. Theor. The bases and altitudes of equal solid parallelepipeds, are reciprocally proportional; and if the bases and altitudes be reciprocally proportional, the solid parallelepipeds are equal.

Prop. XXXV. Theor. If, from the vertices of two equal plane angles, there be drawn two straight lines elevated above the planes in which the angles are, and containing equal angles with the sides of those angles, each to each; and if in the lines above the planes there be taken any points, and from them perpendiculars be drawn to the planes in which the first named angles are: and from the points in which they meet the planes, straight lines be drawn to the vertices of the angles first named; these straight lines shall contain equal angles with the straight lines which are above the planes of the angles.

Prop. XXXVI. Theor. If three straight lines be proportionals, the solid parallelepiped described from all three as its sides, is equal to the equilateral parallelepiped described from the mean proportional, one of the solid angles of which is contained by three plane angles equal, each to each, to the three plane angles containing one of the solid augles of the other figure.

Prop. XXXVII. Theor. If four straight lines be proportionals, the similar solid parallelepipeds similarly described from them shall also be proportionals. And if the similar parallelepipeds simiJarly described from four straight lines be propor tionals, the straight lines shall be proportionals.