« PreviousContinue »
if M has to N the same ratio which A has to D; : 4. That magnitude of which a multiple is grealthen, for shortness sake, M is said to have to N, er than the same multiple of another, is greater the ratio compounded of the ratios of E to F, G than that other magnitude. to H, and K to L
Prop. I. Theor. li' any number of inagnitudes be 12. In proportionals, the antecedent terms are equin ultiples of as many', each of each; what called homologous to one another, as also the multiple soever any one of them is of its part, tho consequents to one anotber.
same multiple shall all the first magnitudes be of Geometers make use of the following technical all the other. words to signify certain ways of changing either Prop. II. Tbeor. If the first magnitude be the the order or magnitude of proportionals, so as that same multiple of the second that the third is of they continue still to be proportionals.
the fourth, and the fifth the same nultiple of the 13. Permutando, or alternando, by permuta. second that the sixth is of the fourth; then shall tion, or alternately; this word is used when there the first together with the fifth be the same mulare four proportionals, and it is inferred, that the tiple of the second, that the third together with first has the sanie ratio to the third, which the the sixth is of the fourth. second has to the fourth; or that the first is to Prop. lII. Theor. If the first be the same multi. the third, as the second to the fourth: as is shewn ple of the second, which the third is of the fourth; ie the XVI. Prop. of this Book V.
and if of the first and third there be taken equi14. Invertendo, by inversion: when there are multiples, these shall be equimultiples, the one of four proportionals, and it is inferred, that the se- the second, and the other of the fourth. cond is to the first, as the fourth to the third. Prop. IV. Theor. If the first of four magnitudes Prop. B. Book V.
has the same ratio to the second which the third 15. Componendo, by composition, when there has to the fourth; then any equiinultiples what. are four proportionals, and it is inferred, that the erer of the first and third shall have the same first, together with the second, is to the sccond, ratio to any equimultiples of the second and as the third together with the fourth, is to the fourth, viz. the equimultiple of the first shall fourth. Prop. XVIII. Book V.
bare the saine ratio to that of the second, which 16. Dividendo, by division; when there are four the equimultiple of the third has to that of the proportionals, and it is inferred, that the excess fourth. of the first above the second, is to the second, as · Prop. V. Theor. If one magnitude be the same the excess of the third above the fourth, is to the multiple of another, which a magnitude taken foartb. Prop. XVII, Book V.
from the first is of a magnitude taken from the 17. Convertendo, by conversion; when there other; the remainder shall be the same multio , are four proportionals, and it is iuferred, that the ple of the remainder, that the whole is of the first is to its excess above the second, as the third whole. to its excess above the fourtb. Prop. E. Book V. Prop. VI. Tbeor, If two magnitudes be equi
18. Ex aequali (se, distantia), or ex aequo, multiples of two others, and if equimultiples of from equality of distance; when there is any nuin- these be taken from the first two, the remainders ber of magnitudes more than two, and as many are either equal to these others, or equimultiples others, so that they are proportionals when taken of them. two and two of each rank, and it is inierred, that Prop. A. Theor. If the first of four magnitudes the first is to the last of the first rank of magni- bas to the second, the same ratio which the third tades, as the first is to the last of the others: of has, to the fourth; then, if the first be greater this there are the two following kinds, which arise than the second, the third is also greater than the from the different order in which the maguitudes fourth; and, if equal, equal; if less, less. are taken two and two.
Prop. B. Theor, If four magnitudes are propor19. Ex aequali, from equality; this term is used tionals, they are proportionals also when taken simply by itself, when the first magnitude is to inversely. the second of the firsi rank, as the first to the Prop. C. Theor. If the first be the same multisecond of the other rank; and as the second is to ple of the second, or the same part of it, that the the third of the first rank, so is the second to the third is of the fourth; the first is to the second, as third of the other; and so on in order, and the in- the third is to the fourth. kerence is as mentioned in the preceding defini- Prop. D. Theor. If the fist be to the second as tion; whence this is called ordinate proportion. It the third to the fourth, and if the first be a multiis demonstrated in Prop. XXII. Book V.
ple, or part of the second; the third is the same 20. Ex aequali, in proportione perturbata, seu inultipie, or the same part of the fourth. inordinata ; from equality, in perturbate or dis- Prop. VII. Theor. Equal magnitudes liave the orderly proportion; this term is used when the same ratio to the same inaguitude; and the same first magnitude is to the second of the first rapk, has the same ratio to equal magnitudes. as the last but one is to the last of the second rank; Prop. VIII. Theor. Of unequal magnitudes, the and as the second is to the third of the first rank, greater has a greater ratio to the same than the 60 is the last but two to the last but one of the less has; and the same magnitude has a greater second rank; and as the third is to the fourth of ratio to the less, than it has to the greater. the first rank, so is the third from the last to the Prop. IX. Theor. Magnitudes which have the last but tuo of the second rank: and so on in a same ratio to the same magnitude are equal ru cross order: and the inference is as in Def. 18. It one another; and those to which the same magni. is demonstrated in Prop. XXIII. Book V.
tude has the same ratio are equal to one allArioms.-1. Equimultiples of the same, or of other. equal magnitudes, are equal to one another. Prop. X. Theor. That magnitude which has a
2. Those magnitudes of which the same, or equal greater ratio thao another bas unto the same magnitudes, are equimultiples, are equal to one magnitude is the greater of the tiro: and that another.
magnitude to which the saine has a greater ratio 3. A multiple of a greater magnitude is greater than it has unto another magnitude is the lesser that the same multiple of a less.
of the two.
Prop. XI. Theor. Ratios that are the same to fourth; and the fifth to the second, the same ratio the same ratio, are the same to one another. which the sixth has to the fourth; the first and
Prop. XII. Thevr. If any number of magnitudes fifth together shall bave to the second, the same be proportionals, as one of the antecedents is to ratio which the third and sixth together have to its consequent, so shall all the antecedents taken the fourth. together be to all the consequents.
Prop. XXV. Theor. If four magnitudes of the Prop. XIII. Theor. If the first has to the second same kind are proportionals, the greatest and least the same ratio which the third has to the fourth, of them together are greater than the other two but the third to the fourth a greater ratio than the together. fifth has to the sixth ; the first shall also have to Prop. F. Theor. Ratios which are compounded the second a greater ratie than the fifth has to the of the same ratios, are the same with one another. sixth.
Prop. G. Theor. If several ratios be the same Prop. XIV. Theor. If the first has to the second, with several ratios, each to each; the ratio wbich the same ratio which the third bas to the fourth; is compounded of ratios which are the same with then, if the first be greater than the third, the the first ratios, each to each, is the same with the second shall be greater than the fourth; and if ratio compounded of ratios which are the same equal, equal; and if less, less.
with the other ratios, each to each. Prop. XV. Theor. Magnitudes have the same Prop. H. Tbeor. If a ratio compounded of seratio to one another which their equimultiples veral ratios be the same with a ratio compounded hare.
of any other ratios, and if one of the first ratios, Prop. XVI. Theor. If four magnitudes of the 'or a ratio compounded of any of the first, be the same kind be proportionals, they shall also be same with one of the last ratios, or with the ratio proportionals when taken alternately.
compounded of any of the last; then the ratio Prop. XVII. Theor. If magnitudes, taken joint- compounded of the remaining ratios of the first, ly, be proportionals, they shall also be proportion- or the remaining ratio of the first, if but one reals when taken separately; that is, if two magni. main, is the same with the ratio compounded of tades together have to one of them the same ratio those remaining of the last, or with the remainwhich two others have to one of these, the re- ing ratio of the last. maining one of the first two shall have to the Prop. K. Theor. If there be any number of raother the same ratio which the remaining one of tios, and any number of other ratios such, that the last two bas to the other of these.
the ratio compounded of ratios which are the same Prob. XVIII. Theor. If magnitudes, taken sepa- with the first ratios, each to each, is the same with rately, be proportionals, they shall also be pro- the ratio eompounded of ratios wbich are the portionals, when taken jointly, that is, if the first same each to each, with the last ratios; and if be to the second, as the third to the fourth, the one of the first ratios, or the ratio which is comfirst and second together shall be to the second, pounded of ratios which are the same with seveas the third and fourth together to the fourth. ral of the first ratios, each to each, be the same
Prop. XIX. Theor. If a whole magnitude be to with one of the last ratios, or with the ratio coma whole, as a magnitude taken from the first, is to pounded of ratios which are the same, each to a magnitude taken from the other; the remainder each, with several of the last ratios: then the shall be to the remainder, as the whole to the ratio compounded of ratios which are the same whole.
with the remaining ratios of the first, each to Prop. E. Theor. Jf four magnitudes be propor- each, or the remaining ratio of the first, if bat tionals, they are also proportionals by conversion, one remain ; is the same with the ratio compoundthat is, the first is to its excess above the second, ed of ratios which are the same with those reas the third to its excess above the fourth.
maining of the last, each to each, or with the rea Prop. XX. Theor. If there be three magnitudes, maining ratio of the last. and other three, which, taken two and two, have Book VI. Def. 1.-Similar rectilineal figures are the same ratio; if the first be greater than the those which have their several angles equal, each third, the fourth shall be greater than the sixth; to each, and the sides about the equal angles proand if equal, equal; and if less, less.
portionals. Prop. XXI. Theor. If there be three magnitudes, 2. Reciprocal figures, viz, triangles and paraland other three, which have the same ratio taken lelograms, are such as have their sides about two two and two, but in a cross order; if the first mag- of their angles proportionals in such manner, that nitude be greater than the third, the fourth shall a side of the first figure is to a side of the other, be greater than the sixth; and if equal, equal; and as the remaining side of this other is to the reif less, less.
maining side of the first. Prop. XXII. Theor. If there be any oumber of 3. A straight line is said to be cut in extreme magnitudes, and as many others, which, taken and mean ratio, when the whole is to the greater two and two in order, have the same ratio; the segment, as the greater segment is to the less. first shall have to the last of the first magnitudes 4. The altitude of any figure is the straight the same ratio which the first of the others has to line drawn from its vertex perpendicular to the the last. N. B. This is usually cited by the words base. ex aequali,” or ex aequo.”
Prop. I. Theor. Triangles and parallelograms of Prop. XXIII. Theor. If there be any number of the same altitude are one to another as their mag.itudes, and as many others, which, taken · bases. two and two, in a cross order, have the same ra- Prop. II. Theor. If a straight line be drawn patio; the first shall have to the last of the first mag- rallel io one of the sides of a triangle, it shall cut nitudes the same ratio which the first of the others the other sides, or those produced, proportionally: has to the last. N. B. This is usually cited by and if the sides, or the sides produced, be cut prothe words, “ ex aequali in proportione perturba- portionally, the straight line which joins the points ta;” or “ ex aequo perturbate,"
of section shall be parallel to the remaining side Prop. XXIV. Theor. If the first has to the se- of the triangle. cond the same ratio which the third has to the Prop. III. Theor. If the angle of a triangle be divided into two equal angles, by a straight line one angle in the one equal to one angle in the #bich also cuts the base; the segments of the base other, and their sides about the equal angles shall bare the same ratio which the other sides of reciprocally proportional, are equal to one anthe triangle have to one another : and if the seg- other. ments of the base have the same ratio which the Prop. XVI. Theor. If four straight lines be pro. other sides of the triangle have to one another, portionals, the rectangle contained by the extremes the straight line drawn from the vertex to the is equal to the rectangle contained by the means : point of section, divides the vertical angle into and if the rectangle contained by the extremes be two equal angles.
equal to the rectangle contained by the means, Prop. A. Theor. If the outward angle of a tri- the four straight lines are proportionals. angle made by producing one of its sides, be di- Prop. XVII. Theor. If three straight lines be rided into two equal angles, by a straight line proportionals, the rectangle contained by the exwhich also cuts the base produced; the segments tremes is equal to the square of the mean: and if between the dividing line and the extremities of the rectangle contained by the extremes be equal the base have the same ratio which the other sides to the square of the mean, the three straight lines of the triangle have to one another: and if the are proportionals. segments of the base produced, have the same Prop. XVIII. Prob. Upon a given straight line ratio which the other sides of the triangle have, to describe a rectilineal figure similar, and simithe straight line drawn from the vertex to the larly situated to a given rectilineal figure. point of section divides the outward angle of the Prop. XIX. Tbeor. Similar triangles are to one triangle into two equal angles.
another in the duplicate ratio of their hoinologous Prop. IV. Theor. The sides about the equal sides. angles of equiangular triangles are proportionals ; Prop. XX. Theor. Similar polygons may be diand those which are opposite to the equal angles vided into the same number of similar triangles, are bomologous sides, that is, are the antecedents baving the same ratio to one another that the poo or consequents of the ratios.
lygons have; and the polygons have to one anProp. V. Theor. If the sides of two triangles, other the duplicate ratio of that which their homo. about each of their angles, be proportionals, the logous sides hare. triangles shall be equiangular, and have their Prop. XXI. Theor. Rectilineal figures which equal angles opposite to the homologous sides. are similar to the same rectilineal figure, are also
Prop. VI. Theor. Iftwo triangles have one angle similar to one another, of the one equal to one angle of the other, and Prop. XXII. Theor, If four straight lines be the sides about the equal angles proportionals, the proportionals, the similar rectilineal figures simi, triangles shall be equiangular, and shall have larly deseribed upon them shall also be proporthose angles equal which are opposite to the ho, tionals; and if the similar rectilincal figures simi. mologous sides.
Jariy described upon four straight lines be proporProp. VII. Theor. If two triangles have one tionals, those straight lines shall be proportionals, angle of the one equal to one angle of the other, Prop. XXIII, Theor. Equiangular paralleloand the sides about two other angles, proportion- grams have to one another the ratio which is coinals, then, if each of the remaining angles be either pounded of the ratios of their sides. less, or not less, than a right angle; or if one of Prop. XXIV, Theor, The parallelograms about them be a right angle: the triangles shall be equi- the diameter of any parallelogram, are similar to angular, and have those angles equal about which the wbole, and to one another. the sides are proportionals.
Prop. XXV. Prob. To describe a rectilineal fi. Prop. VIII. Theor. In a right angled triangle, if gure which shall be similar to one, and equal to a perpendicular be drawn from the right angle to another given rectilineal figure. the base; the triangles on each side of it are simi- Prop. XXVI. Tbeor. If two similar parallelo. lar to the whole triangle, and to one another, grams bave a common angle, and be similarly si-,
Prop. IX. Prob. From a giren strait line to cut tuated; they are about the same diameter, off any part required.
Prop. XXVII. Theor, Of all parallelograms ap; Prop. x. Prob. To divide a given straight line plied to the same straight line, and deficient by similarly to a giveu divided straight line, that is, parallelograms, similar and similarly situated to into parts that shall have the same ratios to one that which is described upon the half of the line;
another which the parts of the divided given that which is applied to the half, and is similar to • straight line bave.
its defect, is the greatest. Prob. XI. Prob. To find a third proportional to Prop. XXVII. Prob, To a given straight line to two given straight lines.
apply a parallelogram equal to a given rectilineal Prop. XII. Prob. To find a fourth proportional figure, and deficient by a parallelogram similar to to three given straight lines.
a given parallelogram; but the given rectilineal Prop. XIII. Prob. To find a mean proportional figure to which the parallelogram to be applied is between two given straight lines.
to be equal, must not be greater than the parallel. Prop. XIV. Theor. Equal parallelograms which ogram applied to half of the given line, having its have one angle of the one equal to one angle of defect similar to the defect of that which is to be the other, have their sides about the equal angles applied; that is, to the given parallelogram. reciprocally proportional: and parallelograms that Prop. XXIX. Prob. To a given straight line to have one angle of the one equal to one angle of apply a parallelogram equal to a given rectilineal the other, and their sides about the equal angles figure, exceeding by a parallelogram similar to reciprocally proportional, are equal to one an- another given. other.
Prop. XXX. Prob. To cut a given straight line Prop. XV. Theor. Equal triangles which have in extreme and mean ratio. one angle of the one equal to one angle of the Prop. XXXI. Theor. In right angled triangles, other, have their sides about the equal angles re- the rectilineal figure described upon the side opciprocally proportional: and triangles which have posite to the right angle, is equal to the similare
and similarly described figures upon the sides con- revolution of a semicircle about its diameter taining the right angle.
which remains unmoved. Prop. XXXII. Theor. If two triangles which 15. The axis of a sphere is the fixed straight have two sides of the one proportional to two line about which the semicircle revolves. sides of the other, be joined at one angle, so as to 16. The centre of a sphere is the same with that have their homologous sides parallel to one an- of the semicircle, other; the remaining sides shall be in a straight 17. The diameter of a sphere is any straight line.
line which passes through the centre, and is terProp. XXXIII. Theor, Io equal circles, angles, minated both ways by the superficies of the whether at the centres or circumferences, have sphere. the same ratio wbich the circumferences on which 18. A cone is a solid figure described by the rethey stand have to one another : so also have the volution of a right angled triangle about one of sectors.
the sides containing the right angle, which side Prop. B. Theor. If an angle of a triangle be bi- remains fixed, sected by a straight line, which likewise cuts the If the fixed side be equal to the other side conbase; the rectangle contained by the sides of the taining the right angle, the coue is called a right triangle is equal to the rectangle contained by the angled cone; if it be less than the other side, an segments of the base, together with the square of obtuse angled, and if greater, an acute angled the straight line bisecting the angle.
Prop. C. Theor. If froin any angle of a triangle a 19. The axis of a cone is the fixed straight line straight line be drawn perpendicular to the base; about which the triangle revolves. the rectaogle contained by the sides of the trian. 20. The base of a cone is the circle described gle is equal to the rectangle contained by the per- by that side containing the right angle, which rependicular and the diameter of the circle describ. volves. ed about the triangle.
21. A cylinder is a solid figure described by the Prop. D. Theor. The rectangle contained by the revolution of a right angled parallelogram about diagonals of a quadrilateral inscribed in a circle, one of its sides which remains fixed. is equal to both the rectangles contained by its 22. The axis of a cylinder is the fixed straight opposite sides.
line about which the parallelogram revolves. Book XI. Def. 1.-A solid is that which hath 23. The bases of a cylinder are the circles de. Jength, breadth, and thickness.
scribed by the two revolving opposite sides of the 2. That which bounds a solid is a superficies. parallelogram.
3. A straight line is perpendicular, or at right 24. Similar cones and cylinders are those wbich angles to a plane, when it makes right angles with have their axes and the diaineters of their bases every straight line meeting it in that plane. proportionals,
4. A plane is perpendicular to a plane, when 25. A cube is a solid figure contained by six the straight lines drawn in one of the planes per- equal squarcs. pendicularly to the cominon section of the two 26. A tetrahedron is a solid figure contained by planes, are perpendicular to the other plage. four equal and equilateral triangles.
5. The inclination of a straight line to a plane 27. An octahedron is a solid figure contained by is the acute ang!e contained by that straight line, eight equal and equilateral triangles. and another drawn from the point in which the 28. A dodecahedron is a solid figure contained first line meets the plane, to the point in which a . by twelve equal pentagons which are equilateral perpendicular to the plane drawn from any point and equiangular, of the first line above the plave, meets the same 29. An icosahedron is a solid figure contained plane.
by twenty equal and equilateral triangles, 6. The inclination of a plane to a plane is the Def. A. A parallelepiped is a solid figure conacute angle contained by two straight lines drawn tained by six quadrilateral figures, whereof every from any the same point of their common section opposite two are parallel. at right angles to it, one upon one plane, and the Prop. I. Theor. One part of a straight line canother upon the other plane.
not be in a plane and another part above it. 7. Two planes are said to have the same, or a Prop. II, Theor. Two straight lines which cut like inclination to one another, which two other one another are in one plane, and three straight planes have, when the said angles of inclination lines which meet one another are in one plane. are equal to one another,
Prop. III. Theor. If two planes cut one another, 8. Parallel planes are such which do not meet their common section is a straight line. one another though produced.
Prop. ĮV. Theur. If a straight line stand at right 9. A solid angle is that which is made by the angles to each of two straight lives in the point of meeting of more than two plapes, which are not their intersection, it shall also be at right angles to in the same plaue, in one point.
the plane which passes through them, that is, to 10. The tenth definition is omitted for reasons the plane in which they are. given in the notes.
Prop. V. Theor. If three straight lines meet all 11. Similar solid figures are such as have all in one point, and a straight line stands at right their solid angles equal, each to each, and which angles to each of them in that point; these three are contained by the same number of similar straight lines are in one and the same plane. planes.
Prop. VI. Theor. If two straight lines be at right 12. A pyramid is a solid figure contained by angles to the same plane, they shall be parallel to planes that are constituted betwixt one plane and one apother. one point above it in which they meet.
Prop. VIị. Theor. If two straight lines be paral. 13. A prism is a sclid figure contained by plane lel, the straight line drawn from any point in the figures of which two that are opposite are equal, one to any point in the other, is in the same plane similar, and parallel to one another; and the with the parallels. others parallelograms.
Prop. VIII. Theor. If two straight lines be pa. 14. A sphere is a solid figure described by the rallel, and one of them is'at right angles to a plane; the other also shall be at right angles to the same opposite planes are similar and equal paralleloplane.
grams. Prop. IX. Theor. Two straight lines which are Prop. XXV. Theor. If a solid parallelepiped be each of them parallel to the same straight line, cut by a plane parallel to two of its opposite planes; and not in the same plane with it, are parallel to it divides the whole into two solids, the base of one one another.
of which shall be to the base of the other, as the Prop. X. Theor. If two straight lines meeting one solid is to the other. one another be parallel to two others that meet Prop. XXVI. Prob. At a given point in a given one another, and are not in the same plane with straight line, to make a solid angle equal to a the first two; the sirst two and the other two shall giveu solid angle contained by three plane augles. contain equal angles.
Prop. XXVII. Prob. To describe from a given Prop. XI. Prob. To draw a straight line perpen- straight line a solid parallelepiped similar, and dicular to a place, from a given point above it. similarly situated to one given.
Prop. XII. Prob. To erect a straight line at right Prop. XXVIII. Theor. If a solid parallelepiped angles to a given plane, from a point given in the be cut by a plane passing through the diagonals plane.
of two of the opposite planes; it shall be cut in Prop. XIII. Theor. From the same point in a two equal parts. given plane, there cannot be two straight lines at Prop. xxix. Theor. Solid parallelepipeds upon right angles to the plane, upon the same side of it: the same base, and of the same altitude, the inand there can be but one perpendicular to a plane sisting straight lines of which are terminated in from a point above the plane.
the same straight lines in the plane opposite to Prop. XIV. Theor. Planes to which the same the base, are equal to one another. straight line is perpendicular, are parallel to one Prop. XXX. Theor. Solid parallelepipeds upon another.
the same base, and of the same altitude, the inProp. XV. Theor. If two straight lines meeting · sisting straight lines of which are not terminated one another, be parallel to two straight lines which in the same straight lines in the plane opposite to meet one another, but are not in the same plane the base, are equal to one another. with the first two; the plane which passes through Prop. XXXI. Theor. Solid parallelepipeds which these is parallel to the plane passing through the are upon cqual bases, and of the same altitude, others.
are equal to one another. Prop. XVI. Theor. If two parallel planes be Prop. XXXII. Theor. Sulid parallelepipeds which cut by another plane, their counmon sectious with have the same altitude, are to one another as their it are parallels.
bases. Prop. XVII. Theor. If two straight lines be cut Prop. XXXIII. Theor, Similar solid paralleleby parallel planes, they sball be cut in the same pipeds are one to another in the triplicate ratio of ratio.
their homologous sides. Prop. XVIII. Theor. If a straight line be at right Prop. D. Theor. Solid parallelepipeds contained angles to a plane, every plane which passes by parallelograms equiangular to one auother, through it shall
be at right angles to that plane. each to each, that is, of which the solid angles are Prop. XIX. Theor. If two planes cutting one equal, each to each, have to one another the ratio another be each of them perpendicular to a third which is the same with the ratio compounded of plane; their common section shall be perpendicu- the ratios of their sides, Iar to the same plane.
Prop. XXXIV. Theor. The bases and altitudes Prop. XX. Theor. If a solid angle be contained of equal solid parallelepipeds, are reciprocally by three plane angles, any two of them are greater proportional; and if the bases and altitudes than tbe third.
be reciprocally proportional, the solid paralleleProp. XXI. Theor. Every solid angle is con- pipeds are equal. tained by plain angles which together are less than Prop. XXXV, Theor. If, from the vertices of four right angles.
two equal plane angles, there be drawn two Prop. XXII. Theor. If every two of three plain straight lines elevated above the planes in wbich angles be greater than the third, and if the straight the angles are, and containing equal angles with lines which contain then be all equal; a triangle the sides of those angles, each to each; and if in may be made of the straight lines that join the the lines above the planes there be taken any extremities of those equal straight lines.
points, and from them perpendiculars be drawn to Prop. XXIII. Prob. To make a solid angle which the planes in wbich the first named angles are: shall be contained by three given plane angles, and from the points in which they meet the planes, any two of them being greater than the third, and straight lines be drawn to the vertices of the anall three together less than four right angles. gles first named; these straight lines shall contain
Prop. A. Theor. If each of two solid angles be equal augles with the straight lines which are contained by three plane angles equal to one an- above the planes of the angles. other, each to each; the planes in which the equal Prop. XXXVI. Theor. If three straight lines be angles are, have the same inclination to one an- proportionals, the solid parallelepiped described other.
from all three as its sides, is equal to the equilaProp. B. Theor. If two solid angles be contain- teral parallelepiped described from the mean proed, each by three plane angles which are equal to portional, one of the solid angles of which is conone another, each to each, and alike situated, these tained by three plane angles equal, each to each, solid angles are equal to one another.
to the three plane angles containing one of the Prop. c. Theor. Solid figures contained by the solid angles of the other figure. same number of equal and similar planes alike Prop. XXXVII. Tbeor. If four straight lines be situated, and having none of their solid angles proportionals, the similar solid parallelepipeds sie contained by more than three plane angles, are milarly described from them shall also be proporequal and similar to one another.
tionals. And if the similar parallelepipeds simiProp. XXIV. Theor. If a solid be contained by Jarly described from four straight lines be propor. six planes, two and two of which are parallel; the tionals, the straight lines shall be proportionals.