D L H K B х As therefore the circumference BC to manner, the sectors EHF, FHM, the circumference EF, so (5. def. 5.) MHN may be proved equal to one is the angle BGC to the angle EHF: another: Therefore, what multiple But as the angle BGC is to the angle soever the circumference BL is of the EHF, so is (15. 5.) the angle BAC to circumference BC, the same multiple the angle EDF; for each is double of is the sector BGL of the sector BGC: each (20. 3.): Therefore, as the cir- for the same reason, whatever mulcumference BC is to EF, so is the tiple the circumference EN is of EF, angle BGC to the angle EHF, and the the same multiple is the sector EHN angle BAC to the angle EDF. of the sector EHF: And if the cir Also, as the circumference BC to cumference BL be equal to EN, the EF, so is the sector BGC to the sector EHF. Join BC, CX, and in the A N M are equal to the two CG, GK, and that they contain equal angles; the base BC is equal (4. 1.) to the base sector BGL is equal to the sector CK, and the triangle GBC to the tri- EHN; and if the circumference BL angle GCK: And because the circum- be greater than EN, the sector BGL ference BC is equal to the circumfer- is greater than the sector EHN; and ence CK, the remaining part of the if less, less: Since then, there are whole circumference of the circle four magnitudes, the two circumfere ABC, is equal to the remaining part ences BC, EF, and the two sectors of the whole circumference of the BGC, EHF, and of the circumference same circle: Wherefore the angle BC, and sector BGC, the circumferBXC is equal to the angle COK (27. ence BL and sector BGL are any 3.); and the segment BXC is there equimultiples whatever; and of the fore similar to the segment COK (11. circumference EF, and sector EHF, def. 3.); and they are upon equal the circumference EN, and sector straight lines BC, CK: But similar EHN, are any equimultiples whatsegments of circles upon equal straight ever; and that it has been proved, if lines, are equal (24. 3.) to one ano- the circumference BL be greater than ther : Therefore the segment BXC is EN, the sector BGL is greater than equal to the segment COK; And the the sector EHN; and if equal, equal; triangle BGC is equal to the triangle and if less, less. Therefore, (5. def. 5.) CGK; therefore the whole, the sec- as the circumference BC is to the cirtor BCG, is equal to the whole, the cumference EF, so is the sector BGC sector CGK: For the same reason, to the sector EHF. Wherefore, in the sector KGL is equal to each of equal circles, &c. Q. E. D. the sectors BGC, CGK: In the same PROP. B. THEOREM. 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. Let ABC be a triangle, and let straight line AD; the rectangle BA, che angle BAC be bisected by the AC is equal to the rectangle BD, B D DC, together with the square of AD. another : Therefore as BA to AD, 90 Describe the cir is (4. 6.) EA to AC, and consequently cle (5.4.)ACB about the rectangle BA, AC is equal (16. 6.) the triangle, and to the rectangle EA, AD, that is, (3. produce AD to the 2.) to the rectangle ED, DA, together circumference in E, with the square of AD: But the rectand join EC: Then angle ED, DA is equal to the rectbecause the angle angle (35.3.) BD, DC. Therefore the BAD is equal to rectangle BA, AC is equal to the rectthe angle CAE, and the angle ABD angle BD, DC, together with the to the angle (21. 3.) AEC, for they square of AD. Wherefore, if an are in the same segment, the triangles angle, &c. Q. E. D ABD, AEC are equiangular to one PROP C. THEOREM. If from an angle of a triangle a straight line be drawn perpendicular to the base; the rectangle contained by the sides of the triangle is equal to the rectangle contained by the perpendicular and the diameter of the circle described about the triangle. Let ABC be a triangle, and AD the and the angle ABD perpendicular from the angle A to the to the apgle AEC in base BC, the rectangle BA, AC is the same segment; equal to the rectangle contained by (31. S.) the triangles AD, and the diameter of the circle ABD, AEC are equidescribed about the triangle. angular: Therefore Describe (5. 4.) the circle ACB a- as (4.6.) BA to AD, bout the triangle, and draw its dia- so is EA to AC; and meter AE, and join EC: Because the consequently the rectangle BA, AC is right angle BĎA is equal (31. 3.) equal (16. 6.) to the rectangle EA, AD. to the angle ECA in a semicircle, if therefore from an angle, &c. Q. E. D. B PROB. D. THEOREM. B The rectangle contained by the diagonals of a quadrilateral inscribed in a circle, is equal to both the rectangles contained by its opposite sides. Let ABCD be any quadrilateral in- the angle BCE, scribed in a circle, and join AC, BD; because they are the rectangle contained by AC, BD is in the same segequal to the two rectangles contained ment ; therefore by AB, CD, and by AD, BC. the triangle ABD Make the angle ABE equal to the is eqniangular to angle DBC , add to each of these the the triangle BCE: common angle EBD, then the angle Wherefore, (4. 6.) ABD is equal to the angle EBC; and as BC is to CE so is BD to DA; and the angle BDA is equal (21. 3.) to consequently the rectangle BC, AD is A • This is a Lenama of Ptolomæus, in page 9 of his pasyann aurratus. equal(16.6.) to the rectangle BD, CE: AE: But the rectangle BC, AD has Again, because the angle ABE is equal been shewn equal to the rectangle BD, to the angle DBC, and the angle (21. CE; therefore the whole rectangle AC, 3.) BAE to the angle BDC, the triangle BD (4. 6.) is equal to the rectangle ARE is equiangular to the triangle AB, DC, together with the rectangle BCD: As therefore BA to AE, so is AD, BC. Therefore the rectangle, BD to DC; wherefore the rectangle &c. Q. E. D. BA, DC is equal to the rectangle BD, BOOK XI. DEFINITIONS. I. at right angles to it, one upon one A solid is that which hath length, plane, and the other upon the other breadth, and thickness. plane. II. VII. That which bounds a solid is a su- Two planes are said to have the same, perficies. er a like inclination to one another, III. which two other planes have, when A straight line is perpendicular, or at the said angles of inclination are right angles to a plane, when it equal to one another. makes right angles with every VIII. straight line meeting it in that Parallel planes are such which do not plane. meet one another though produced. IV. IX. A plane is perpendicular to a plane, A solid angle is that which is made by when the straight lines drawn in the meeting of more than two plane one of the planes perpendicularly angles, which are not in the same to the common section of the two plane, in one point. planes, are perpendicular to the X. other plane. · The tenth definition is omitted for V. reasons given in the notes.' The inclination of a straight line to a XI. plane is the acute angle contained Similar solid figures are such as have by that straight line, and another all their solid angles equal each to drawn from the point in which the each, and which are contained by first line meets the plane, to the the same number of similar planes. point in which a perpendicular to XII. the plane, drawn from any point of A pyramid is a solid figure contained the first line above the plane, meets by planes that are constituted bethe same plane. twixt one plane and one point above VI. it in which they meet. The inclination of a plane to a plane XIII. is the acute angle contained by two A prism is a solid figure contained by straight lines drawn from any the plane figures, of which two that are same point of their common section opposite are equal, similiar, and pa rallel to one another; and the others by the revolution of a right angled parallelograms. parallelogram about one of its sides XIV. which remains fixed. A sphere is a solid figure described XXII. by the revolution of a semicircle The axis of a cylinder is the fixed about its diameter, which remains straight line about which the paunmoved. rallelogram revolves. XV. XXIII. The axis of a sphere is the fixed The bases of a cylinder are the circles straight line about which the semi- described by the two revolving ope circle revolves. posite sides of the parallelogram. XVI. XIV. The centre of a sphere is the same Similar cones and cylinders are those with that of the semicircle. which have their axes and the diaXVII. meters of their bases proportionals. The diameter of a sphere is any XXV. straight line which passes through A cube is a solid figure contained by the centre, ard is terminated both six equal squares. ways by the superficies of the sphere. XXVI. A tetrahedron is a solid figure conA cone is a solid figure described by tained by four equal and equilateral the revolution of a right angled tri- triangles. angle about one of the sides con XXVII. taining the right angle, which side An octahedron is a solid figura conremains fixed. tained by eight equal and equilaIf the fixed side be equal to the other teral triangles. side containing the right angle, the XXVIII. cone is called a right angled cone; A dodecahedron is a solid figrire conif it be less than the other side, an tained by twelve equal pentagons obtuse angled, and if greater, an which are equilateral and equianacute angled cone. gular. XIX. XXIX. The axis of a cone is the fixed straight An icosahedron is a solid figure con line about which the triangle re- tained by twenty equal and equivolves. lateral triangles. XX. DEF. A. The base of a cone is the circle de- A parallelopiped is a solid figure con scribed by that side containing the tained by six quadrilateral figures, right angle, which revolves. whereof every opposite , two are XXI. parallel. A cylinder is a solid figure described PROP. I. THEOREM. One part of a straight line cannot be in a plane, and another part above it. If it be possible, let AB, part of the be produced to D: straight line ABC, be in the plane, And let any plane and the part BC above it : And since pass through the the straight line AB is in the plane, it straight line AD, A B D can be produced in that plane: Let it and be turned about it until it pass through the point C; the same plane that have a common and because the points B, C are in segment AB, which is impossible. this plane, the straight line BC is in (Cor. 11. 1.) Therefore, one part, &c. it (1. Def. 1.): Therefore there Q. E. D. are two straight lines ABC, ABD in PROP. II. THEOREM. Two straight lines which cut one another are in one plane, and three straight lines which meet one another are in one plane. Let two straight lines AB, CD, cut the straight line BC is in A D one another in E; AB, CD are in one the same; and, by the plane: _And three straight lines EC, hypothesis, EB is in it: CB, BE, which meet one another, are Therefore the three in one plane. straight lines EC, CB, Let any plane pass through the BE are in one plane : But straight line EB, and let the plane be in the plane in which EC, turned about EB, produced, if neces- EB are, in the same areC sary, until it pass through the point (1. 11.) CD, AB: Therefore AB, CD, C: Then because the points E, C are are in one plane. Wherefore two in this plane, the straight line EC is straight lines, &c. Q. E. D. in it (7. Def. 1.): For the same reason PROP. III. THEOREM. If two planes cut one another, their common seclion is a straight line. Let two planes AB, BC, cut one in the plane BC, the straight line another, and let the DFB: Then two straight lines DEB, line DB be their B в DFB have the same extremities, and section : therefore include a space betwixt DB is a straight them: which is impossible (10. Ax. 1.): line : If it be not, Therefore BD the common section of from the point D the planes AB, BC cannot but be a to B, draw, in the straight line. Wherefore, if two plane AB, the straight line DEB, and planes, &c. Q. E. D. common PROP. IV. THEOREM. 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: Let the straight line EF stand at ED all equal to one another; and right angles to each of the straight through E draw, in the plane in which lines AB, CD in E, the point of their are AB, CD, any straight line GEH; intersection: EF is also at right angles and join AD, CB; then from any to the plane passing through AB, CD. point F in EF, draw FA, FG, FD, Take the straight lines AE, EB, CE, FC, FH, FB: And because the two |