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has been done upon BK, and the the circle; therefore the circumferike be done also in the other three ence subtended by KB is greater than quadrants, and in the other hemi- the fourth part of the whole circum. sphere; there shall be formed a solid ference of the circle KBOS, and conpolyhedron described in the sphere, sequently the angle BZK at the centre composed of pyramids the bases of is greater than a right angle: And which are the aforesaid quadrilateral because the angle BZK is obtuse, the figures, and the triangle YRX, and square of BK is greater (12. 2.) than and those formed in the like manner the squares of BZ, ZK; that is greater in the rest of the sphere, the common than twice the square of BZ. Join vertex of them all being the point A: KV, and because in the triangles KBV, And the superficies of this solid poly- OBV, KB, BV are equal to OB, BV, hedron does not meet the lesser sphere and that they contain equal angles ; in which is the circle FGH: For, from the angle KVB is equal (4. 1.) to the the point A draw (11. 11.) AZ perpen- angle OVB : and OÛB is a right andicular to the plane of the quadrila- gle; therefore also KVB is a right teral KBOS, meeting it in 2, and join angle: And because BD is lese than BZ, ZK: And because AZ is perpen- twice DV, the rectangle contained by dicular to the plane KBOS, it makes DB, BV is less than twice the rectright angles with every straight line angle DVB; that is, (8. 6.) the square meeting it in that plane; therefore AZ of KB is less than twice the square of is perpendicular to BZ and ZK: And KV: But the square of KB is greater because AB is equal to AK, and that than twice the square of BZ; therethe squares of AZ, ZB, fare equal to fore the square of KV is greater than the square of AB; and the squares of the square of BZ: And because BA is AZ, ZK to the square of AK (47.1.); equal to AK, and that the squares of therefore the squares of AZ, ZB are BZ, ZA are equal together to the equal to the squares of AZ, ZK: Take square of BA, and the squares of KV, from these equals the square of AZ; VA to the square of AK; therefore the remaining square of BZ is equal the squares of BZ, ZA are equal 10 to the remaining square of ZK; and the squares of KV, VA; and of these therefore the straight line BZ is equal the square of KV is greater than the to ZK: In the like manner it may be square of BZ, therefore the square of demonstrated, that the straight lines VA is less than the square of ZA, and drawn from the point Z to the points the straight line AZ greater than VA: O, S are equal to BZ or ZK: There- Much more then is AZ greater than fore the circle described from the cen- AG; because, in the preceding protre 2, and distance ŽB, shall pass position, it was shown that KV falls through the points K, 0, S, and KBOS without the circle FGH; and AZ is shall be a quadrilateral figure in the perpendicular to the plane KBOS, circle: And because KB is greater and is therefore the shortest of all the than QV, and QV equal to So, there- straight lines that can be drawn from fore KB is greater than SO: But KB A, the centre of the sphere to that is equal to each of the straight lines plane. Therefore the plane KBOS BO, KS; wherefore each of the cir- does not meet the lesser sphere. cumferences cut off by KB, BO, KS And that the other planes between is greater than that cut off by Os; the quadrants BX, KX fall without and these three circumferences, toge. the lesser sphere, is thus demonstratther with a fourth equal to one of ed: From the point A draw AI perthem, are greater than the same three pendicular to the plane of the quadritogether with that cut off by OS ; that lateral SOPT, and join 10; and, as is, than the whole circumference of was demonstrated of the plane KBUS
and the point 'Z, in the same way it straight line equal to GU, inscribed may be shown that the point I is the in the circle BCDE: Let this be the centre of a circle described about circumference KB: Therefore the SOPT: and that OS is greater than straight line KB is less than GU : PT; and PT was shown to be pa And because the angle BZK is obrallel to OS: Therefore, because the tuse, as was proved in the preceding, two trapeziums KBOS, SOPT in
therefore BK is greater than BZ: scribed in circles have their sides BK, But GU is greater than BK; much OS parallel, as also OS, PT; and more then is GU greater than BZ, their
other sides BO, KS, OP, ST all and the square of Gų than the square equal to one another, and that BK is of BZ ; and AU is equal to AB; greater than OS, and OS greater than therefore the square of AU, that is, PT, therefore the straight line ZB is the squares of AG, GU, are equal to greater (2. Lem. 12.) than IO. Join the square of AB, that is to the AO which will be equal to AB; and squares of AZ, ZB; but the square of because AIO, AZB, are right angles, BZ is less than the square of GU ; the squares of AI, 10 are equal to the therefore the square of AZ is greater square of A0 or of AB; that is, to the than the square of AG, and the straight squares of AZ, ZB; and the square of line AZ consequently greater than the ZB is greater than the square of 10, straight line AG. therefore the square of AZ is less than Cor. And if in the lesser sphere the square of Al; and the straight there be described a solid polyhedron, line AZ less than the straight line by drawing straight lines betwixt the AI: And it was proved, that AZ is points in which the straight lines greater than AG; much
more then is from the centre of the sphere drawn AI greater than AG: Therefore the to all the angles of the solid polyplane SOPT falls wholly without the hedron in the greater sphere meet jesser sphere: In the same manner it the superficies of the lesser ; in the may be demonstrated, that the plane same order in which are joined the TPRY falls without the same sphere, points in which the same iines from as also the triangle YRX, viz. by the the centre meet the superficies of the Cor. of 2d. Lemma. And after the greater sphere; the solid polyhedron same way it may be demonstrated, in the sphere BCDE has to this other that all the planes which contain the solid polyhedron the triplicate ratio of solid polyhedron, fall without the that which the diameter of the sphere lesser sphere. Therefore in the greater BCDE has to the diameter of the of two spheres, which have the same other sphere: For if these two solids centre, a solid polyhedron is described, be divided into the same number of the superficies of which does not meet pyramids, and in the same order, the the lesser sphere. Which was to be pyramids shall be similar to one ano. done.
ther, each to each: Because they have But the straight line AZ may be the solid angles at their common verdemonstrated to be greater than AG tex, the centre of the sphere, the same otherwise, and in a shorter manner, in each pyramid, and their other solid without the help of Prop. 16, as fol- angle at the bases equal to one anolows. From the point G draw GU ther, each to each, (B. 1.) because at right angles to AG, and join AU. they are contained by three plane anIf then the circumference BE be bio gles, equal each to each; and the py. bected, and its half again bisected, and ramids are contained by the same so on, there will at length be left a number of similar planes; and are circumference less than the circum- therefore similar (il. def. 11.) to ference which is subtended by a one another, each to each :) Bu
similar pyramids have to one ano each of the same order in the lesser, ther the triplicate (2. Cor. 8. 12.) the triplicate ratio of that which AB ratio of their homologous sides : has to the semidiameter of the lesser Tterefore the pyramid of which the sphere. And as one antecedent is to base is the quadrilateral KBOS, and its consequent, so are all the antecevertex A, has to the pyramid in the dents to all the consequents. Whereother sphere of the same order, the fore the whole solid polyhedron in the triplicate ratio of their homologous greater sphere has to the whole solid sides ; that is, of that ratio which AB polyhedron in the other, the triplicate from the centre of the greater sphere ratio of that which AB the semidihas to the straight line from the same ameter of the first has to the semicentre to the superficies of the lesser diameter of the other; that is, which sphere. And in like manner, each the diameter BD of the greater has to pyramid in the greater sphere has to the diameter of the other sphere.
PROBLEM XVIII. THEOREM.
Spheres have to one another the triplicate ratio of that which their dia.
Let ABC, DEF be two spheres, of the sphere ABC describe another which the diameters are BC, EF. similar to that in the sphere DEF: The sphere ABC has to the sphere Therefore the solid polyhedron in the DEF the triplicate ratio of that which sphere ABC has to the solid polyhe. BC has to EF.
dron in the sphere DEF, the triplicate For, if it has not, the sphere ABC ratio (Cor. 17. 12.) of that which BC shall have to a sphere either less or has to EF. But the sphere ABC has greater than DEF, the triplicate ratio to?e sphere GHK, the triplicate ratio of that which BC has to EF. First, of that which BC has to EF; there let it have that ratio to a less, viz. to fore as the sphere ABC to the sphere the sphere GHK; and let the sphere GHK, so is the said polyhedron in the DEF have the same centre with sphere ABC to the solid polyhedron GHK; and in the greater sphere DEF in the sphere DEF: But the sphere describe (17. 12.) a solid polyhedron, ABC is greater than the solid polyhethe superficies of which does not dron in it; therefore (14.5.) also the meet the lesser sphere GHIK; and in sphere GHK is greater than the solid
polyhedron in the sphere DEF: But strated, that the sphere DEF has not it is also less, because it is contained to any sphere less than ABC, the within it, which is ir possil.le : There- triplicate ratio of that which EF has fore the spbero ABC has not to any to BC. Nor can the sphere ABC sphere less than DEF, the triplicate have to any sphere greater than DEF, ratio cf ihat which BC käs to EF. In the triplicate ratio of that which BC the same manner i: may be demon- has to EF: For, if it can, let it have
that ratio to a greater sphere LMN: less than ABC the triplicate ratio of Therefore, by inversion, the sphere that which EF has to BC; which was LMN has to the sphere ABC, the shown to be impossible: Therefore triplicate ratio of that which the the sphere ABC has not to any sphere diameter EF has to the diameter BC. greater than DEF the triplicate ratio But as the sphere LMN to ABC so is of that which BC has to EF: And it the sphere DEF to some sphere, which was demonstrated, that neither has it must be less (14. 5.) than the sphere that ratio to any sphere less than ABC, because the sphere LMN is DEF. Therefore the sphere ABC has greater than the sphere DEF: There. to the sphere DEF, the triplicate ratio fore the sphere DEF bas to a sphere of that which BC has to EF. Q. E. D.
The uses of Plane Geometry are too numerous to be inserted in this place; we may, however, mention a few. Every branch of mathematics which regards lines, surfaces, and solids, are entirely dependent on its principles. Mensuration, and the whole of that curious and entertaining branch of science called Trigonometry, is nothing but the application of Geometry; the Conic Sections cannot be established without a knowledge of its elements. It is indispensible in navigation, geography, astronomy, projection, projection of the sphere, perspective, dialing, &c.
There is no mechanical profession that does not derive considerable advantage from it; and even one workman in the line of his profession, is as much superior to another as he understands more of geometry. By its means the architect lays down his plans, and erects his edifice, and the engineer directs canals to be cut, and bridges to be erected over large rivers.
In short it is impossible to view the landscape before us with advantage, or to give a just account of things which we see, while we are ignorant of Geometry.
TRIGONOMETRY is that part of mathematics which teaches us to investigate the relations that obtain between the sides and angles of triangles.
It is usually divided into two branches.
PLANE TRIGONOMETRY, which determines the relation between the parts of plane triangles, or triangles formed by right lines only; and
SPHERICAL TRIGONOMETRY, which concerns triangles formed by arcs of circles.
Both are of very great importance; for on one or either of them depend the theories of surveying, astronomy, navigation, dialling, and many other branches of mixed mathematics.
We shall commence with plane Trigonometry, and as a preliminary step, explain the Theory of Sines.
THEORY AND ARITHMETIC OF SINES.
1. The sine of an arc or of an angle, is a line which, proceeding from one of the ex
I tremities of the arc or angle, falls perpendicularly on the radius or diameter, passing through the other extremity. Thus the perpendicular BD, drawn from the extremity B, of the arc BA, upon the radius CA, which passes through the other extremity, is called the sinc of the arc AB, or of the angle ACB measured by that arc.
2. If the arc EB= 90--arc BA, or in other words, if the are EB is the complement of the arc BA, its sine GB is the sine of the complement, or the cosine of the arc AB. It is evident that CD= BG, and that BD=GC.
3. The perpendicular AT, drawn from the extremity A, of the radius CA, till it meets with the radius CB prolonged, is called the tangent of the arc AB, and CT is the secant of that arc. Similarly the tangent EM of the arc EB is the tangent of the complement or the cotangent of the arc AB; and CM is the cosecant.
The lines AD, EG are called the versed sine and coversed sine : they are seldom used.
By way of abbreviation we shall write sin, cos, tang, cot, sec, cosec, versin, coversin, instead of sine, cosine, tangent, cotangent, secant, cosecant, versed sine, and co-versed sine.
4. From the definition of the sine, it follows that sine of any arc