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Let ABC, DEF be two spheres, of which the diameters are BC, EF. The sphere ABC shall have to the sphere DEF the triplicate ratio
of that which BC has to EF.
For, if it has not, the sphere ABC must have to a sphere either less or greater than DEF, the triplicate ratio of that which BC has to EF. First, if possible, let it have that ratio to a less, viz. to the sphere GHK;
and let the sphere DEF have the same centre with ĜHK: and in the greater sphere DEF inscribe a solid polyhedron, the super
ficies of which does not meet the lesser sphere GHK; (XII. 17.) and in the sphere ABC inscribe another similar to that in the
sphere DÈF: therefore the solid polyhedron in the sphere ABC has to the solid polyhedron in the sphere DEF, the triplicate ratio of that which BC has to EF. (XII. 17. Cor.) But the sphere ABC has to the sphere GHK, the triplicate ratio of
that which BC has to EF; therefore, as the sphere ABC to the sphere GHK, so is the solid polyhedron in the sphere ABC to the solid polyhedron in the sphere DÉF:
but the sphere ABC is greater than the solid polyhedron in it; therefore also the sphere GHK is greater than the solid polyhedron
in the sphere DEF: (v. 14.) but it is also less, because it is contained within it, which is impossible: therefore the sphere ABC has not to any sphere less than DEF,
the triplicate ratio of that which BC has to EF. In the same manner, it may be demonstrated, that the sphere DEF has not to any sphere less than ABC, the triplicate ratio of that which EF has to BC. Nor can the sphere ABC have to any sphere greater than DEF, the
triplicate ratio of that which BC has to EF: for, if it can, let it have that ratio to a greater sphere LMN: therefore, by inversion, the sphere LMN has to the sphere ABC, the triplicate ratio of that which the diameter EF has to the diameter BC. But as the sphere LMN to ABC, so is the sphere DEF to some
sphere, which must be less than the sphere ABC, (v. 14.) because the sphere LMN is greater than the sphere DEF; therefore the sphere DEF has to a sphere less than ABC the triplicate ratio of that which EF has to BC;
which was shewn to be impossible : therefore the sphere ABC has not to any sphere greater than DEF,
the triplicate ratio of that which BC has to ĚF: and it was demonstrated, that neither has it that ratio to any sphere
less than DEF. Therefore the sphere ABC has to the sphere DEF, the triplicate
ratio of that which BC has to EF. Q. E.D.
NOTES TO BOOK XII.
This book treats of the properties of prisms and cylinders, pyramids and cones. A new principle is introduced called “the method of Exhaustions,” which may be applied for the purpose of finding the areas and ratios of circles, and the relations of the surfaces and of the volumes of cones, spheres and cylinders.
The first comparison of rectilinear areas is made in the first book of the Elements by the principle of superposition, where two triangles are coincident in all respects ; next, comparison is made between triangles and other rectilinear figures when they are not coincident.
In the sixth book, similar triangles are compared by shewing that they are in the duplicate ratio of their homologous sides, and then by dividing similar polygons into the same number of similar triangles, and shewing that the polygons are also in the duplicate ratio of any of their homologous sides. In the eleventh book, similar rectilinear solids are compared by shewing that their volumes are to one another in the triplicate ratio of their homologous sides.
“ The method of Exhaustions” is founded on the principle of exhausting a magnitude by continually taking away a certain part of it, as is explained in the tenth book of the Elements, where Euclid states, that two quantities are equal, whose difference is less than any assignable quantity. If A and A' be two magnitudes of the same kind, and if d and ď' be any other magnitudes of the same kind, such that A'+ d is always greater than, and A' – ďalways less than A, however small d and d'
may be made, then A' is equal to A.
The method of exhaustions may be applied to find the circumference and area of a circle. A rectilinear figure may be inscribed in the circle and a similar one circumscribed about it, and then by continually doubling the number of sides of the inscribed and circumscribed polygons, by this principle, it may be demonstrated, that the area of the circle is less than the area of the circumscribed polygon, but greater than the area of the inscribed polygon; and that as the number of sides of the polygon is in. creased, and consequently the magnitude of each diminished, the differences between the circle and the inscribed and circumscribed polygons are continually exhausted, and at length become less than any assignable difference ; and the area of the circle is the limit to which the inscribed and circumscribed polygons continually approach, as the number of sides is increased.
Also in comparing two unequal circles, two similar polygons are inscribed in the circles, and then by doubling the number of sides continually, it is shewn that the limit to which the ratio of the areas of the rectilineal figures continually approach is the same as the ratio of the circles.
In a similar way it will be seen, that the principle is applied to the surfaces and volumes of cones, cylinders and spheres.
Prop. 11. (1.) For there is some square equal to the circle ABCD; let P be the side of it, and to three straight lines BD, FH, and P, there can be a fourth proportional ; let this be Q: therefore (vi. 22.) the squares of these four straight lines are proportionals; that is, to the squares of BD, FH, and the circle ABCD it is possible there may be a fourth proportional. Let this be S. And in like manner are to be understood some things in some of the following propositions.
(2.) For as, in the foregoing note, it was explained how it was possible there could be a fourth proportional to the squares of BD, FH, and the circle ABCD, which was named S; so, in like manner, there can be a fourth proportional to this other space, named T, and the circles ABCD, EFGH. And the like is to be understood in some of the following propositions.
(3.) Because, as a fourth proportional to the squares of BD, FH, and the circle ABCD, is possible, and that it can neither be less nor greater than the circle EFGH, it must be equal to it. Simson.
Prop. iv. Because GO is equal to OA, and GM to MB, therefore (v1. 2.) OM is parallel to AB; in the same manner ON is parallel to AC; therefore (X1. 15.) the plane MON is parallel to the plane BAC. Simson.
Prop. XI. Vertex is put in place of altitude, which is in the Greek, because the pyramid, in what follows, is supposed to be circumscribed about the cone, and so must have the same vertex. And the same change is made in some places following. Simson.
The thirteenth book of the Elements relates to equilateral and equiangular figures inscribed in circles, and to the five regular solids.
Two books which treat of the inscriptions of the five regular solids in one another and in spheres, are frequently found annexed to the Elements as the fifteenth and six. teenth books ; these however, were composed by Hypsicles of Alexandria.
There is a continuation of the same subject by Flussas, which has been appended to the Elements, and called the sixteenth book of the Elements,
ANCIENT GEOMETRICAL ANALYSIS.
SYNTHESIS, or the method of composition, is a mode of reasoning which begins with something given, and ends with something required, either to be done or to be proved. This may be termed a direct process, as it leads from principles to consequences.
Analysis, or the method of resolution, is the reverse of synthesis, and thus may be considered an indirect process, a method of reasoning from consequences to principles.
The synthetic method is pursued by Euclid in his elements of Geometry. He commences with certain assumed principles, and proceeds to the solution of problems and the demonstration of theorems by undeniable and successive inferences from them.
The Geometrical Analysis was a process employed by the ancient Geometers, both for the discovery of the solution of problems and for the investigation of the truth of theorems. In the analysis of a problem, the quæsita, or what is required to be done, is supposed to have been effected, and the consequences are traced by a series of geometrical constructions and reasonings, till at length they terminate in the data of the problem, or in some previously demonstrated or admitted truth, whence the direct solution of the problem is deduced.
In the Synthesis of a problem, however, the last consequence of the analysis is assumed as the first step of the process, and by proceeding in a contrary order through the several steps of the analysis until the process terminate in the quæsita, the solution of the problem is effected.
But if, in the analysis, we arrive at a consequence which contradicts any truth demonstrated in the Elements, or which is inconsistent with the data of the problem, the problem must be impossible: and further, if in certain relations of the given magnitudes the construction be possible, while in other relations it is impossible, the discovery of these relations will become a necessary part of the solution of the problem.
In the analysis of a theorem, the question to be determined, is, whether by the application of the geometrical truths proved in the Elements, the predicate is consistent with the hypothesis. This point is ascertained by assuming the predicate to be true, and by deducing the successive consequences of this assumption combined with proved geometrical truths, till they terminate in the hypothesis of the theorem or some demonstrated truth. The theorem will be proved synthetically by retracing, in order, the steps of the investigation pursued in the analysis, till they terminate in the predicate, which was assumed in the analysis. This process will constitute the demonstration of the theorem.
If the assumption of the truth of the predicate in the analysis lead to some consequence which is inconsistent with any demonstrated truth,
the false conclusion thus arrived at indicates the falsehood of the predicate; and by reversing the process of the analysis, it may be demonstrated, that the theorem cannot be true.
It may be here remarked, that the geometrical analysis is more extensively useful in discovering the solution of problems than for investigating the demonstration of theorems.
From the nature of the subject, it must be at once obvious, that no general rules can be prescribed, which will be found applicable in all cases, and infallibly lead to the solution of every problem. The conditions of problems must suggest what constructions may be possible; and the consequences which follow from these constructions and the assumed solution, will shew the possibility or impossibility of arriving at some known property consistent with the data of the problem.
Though the data of a problem may be given in magnitude and position, certain ambiguities will arise, if they are not properly restricted. Two points may be considered as situated on the same side, or one on each side of a given line; and there may be two lines drawn from a given point making equal angles with a line given in position; and to avoid ambiguity, it must be stated on which side of the line the angle is to be formed.
A problem is said to be determinate when, with the prescribed conditions, it admits of one definite solution; the same construction which may be made on the other side of any given line, not being considered a different solution: and a problem is said to be indeterminate when it admits of more than one definite solution. This latter circumstance arises from the data not absolutely fixing but merely restricting the quæsita, leaving certain points or lines not fixed in one position only. The number of given conditions may be insufficient for a single determinate solution; or relations may subsist among some of the given conditions from which one or more of the remaining given conditions may be deduced.
If the base of a right-angled triangle be given, and also the difference of the squares of the hypothenuse and perpendicular, the triangle is indeterminate. For though apparently here are three things given, the right angle, the base, and the difference of the squares of the hypothenuse and perpendicular, it is obvious that these three apparent conditions are in fact reducible to two: for since in a right-angled triangle, the sum of the squares on the base and on the perpendicular are equal to the square on the hypothenuse, it follows that the difference of the squares of the hypothenuse and perpendicular is equal to the square of the base of the triangle, and therefore the base is known from the difference of the squares of the hypothenuse and perpendicular being known. The conditions therefore are insufficient to determine a right-angled triangle; an indefinite number of triangles may be found with the prescribed conditions, whose vertices will lie in the line which is perpendicular to the base.
If a problem relate to the determination of a single point, and the data be sufficient to determine the position of that point, the problem is determinate: but if one or more of the conditions be omitted, the data which remain may be sufficient for the determination of more than one point, each of which satisfies the conditions of the problem ; in that case, the problem is indeterminate, and in general such points are found