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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 part of it, as it is explained in the first Lemma of the Tenth Book of the Elements.
"The method of Exhaustions" was employed by the Ancient Geometers and was strictly rigorous in its principles; but it was too tedious and operose in its application to be of extensive utility as an instrument of investigation. It is exemplified in Euc. XII. 2, where it is proved that the areas of circles are proportional to the squares on their diameters. In demonstrating this truth, it is first shewn by inscribing successively in one of the circles regular polygons of four, eight, sixteen, &c. sides, and thus tending to exhaust the area of the circle, that a polygon may be found which shall differ from the circle by a quantity less than any magnitude which can be assigned: and then since similar polygons inscribed in the circles are as the squares on their diameters (Euc. XII. 1.) the truth of the proposition is established by means of an indirect proof.
"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 increased, and consequently the magnitude of each diminished, the differences between the circle and the inscribed and circumscribed polygons are continually exhausted.
In a similar way, the principle is applied to the surfaces and volumes of cones, cylinders and spheres.
As only the first and second propositions of the Twelfth Book of the Elements are required to be read for Honours at Cambridge, the demonstrations of the remaining fifteen propositions of this book have been omitted. The second proposition is perhaps retained merely as an example of the method employed by the ancient Geometers. This method has been replaced by that of prime and ultimate ratios, which is now employed in the proofs of such propositions as were formerly effected by "the method of Exhaustions.”
ON THE PROPOSITIONS IN THE ELEMENTS.
THERE are only two forms of Propositions in the Elements, the theorem and the problem. In the theorem, it is asserted, and is to be proved, that if a geometrical figure be constructed with certain specified conditions, then some other specified relations must necessarily exist between the constituent parts of that figure. Thus: if squares be described on the sides and hypotenuse of a right-angled triangle, the square on the hypotenuse must necessarily be equal to the other two squares upon the sides (Euc. 1. 47). In the problem, certain things are given in magnitude, position, or both, and it is required to find certain other things in magnitude, position, or both, that shall nevessarily have a specified relation to the things given, or to each other, or to both of them. Thus :-a circle being given, it may be required to construct a pentagon, which shall have its angular points in the circumference, and which shall also have both all its sides equal, and all its angles equal. (Euc. iv. 11.)
A problem is said to be determinate, when with the prescribed conditions it admits of one definite solution: and it 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 than one of the remaining given conditions may be deduced.
It may be remarked in Euclid's propositions, that there is in general, an aim at definiteness, considered in reference to the quæsitum of the problem, and the predicate of the theorem. The quæsitum of the problem is either a single thing, as the perpendicular in Euc. 1. 11; or at most, two, as the tangents to the circle in the first case of Euc. III. 17; and in the most general problems, even those which transcend the ordinary geometry, the solutions are, in general, restricted to a definite number, which can always be assigned a priori for every problem. In certain cases, however, the conditions given in Euclid are not sufficient to fix entirely the quæsitum in all respects. For instance, in Euc. 1. 2, it has been seen that the position of the line required is not fixed by the conditions of the problem, so that more lines than one can be drawn from the given point fulfilling the required conditions: nor is the direction prescribed in which the line is to be drawn, so that it has been seen, that from the given point, two lines in opposite directions can be drawn from the given point for each of the possible constructions. And in Euc. Iv. 10, the magnitude of the triangle is any whatever, and therefore not entirely fixed in all respects: or, again, in Euclid Iv. 11, the pentagon may be any whatever, so that its position in the circle is not fixed. To fix the magnitude of the triangle, or the position of the pentagon, some other condition independent of the data, must be added to the conditions of the problem. The length and position of some line connected with the triangle, (as one of the equal sides, the base, the perpendicular, &c.) would have fixed the triangle in magnitude and position; and the position of one angular point of the pentagon, or the condition that one side of the pentagon should pass
through a given point (though this point must be subject to a certain restriction as to position, if within the circle), or any other possible conditions, would have confined the pentagon to a single position, or to the alternative of two positions. Such is the only kind of indeterminateness in the problems of "the Elements." In the enunciation of the theorems too, the same aim at singleness in the property asserted to be consequent on the hypothesis, is apparent throughout. There is, however, a remarkable difference in the characters of the hypotheses themselves, in Euclid's theorems: viz.
1. That in some of them, one thing alone, or a certain definite number, possesses the property which is affirmed in the enunciation. 2. That in others, all the things constituted subject to the hypothetical conditions, possess the affirmed property.
As instances of the first class, the greater number of theorems in the Elements may be referred to, as Euc. I. 4, 5, 6, 8, which are of the simplest class. In these, only one thing is asserted to be equal to another specified thing. In all the theorems of the Second Book, one thing is asserted to be equal to several other things taken together; and the same occurs in Euc. 1. 47, as well as frequently in the other Books. They sometimes also take the form of asserting that no certain magnitude is greater or less than another, as in Euc. 1. 16, or that two things together are less than, or greater than, some one thing or several things, as Euc. 1. 17. In all cases, however, this class is distinguished by the circumstance, that the things asserted to have the property are of a given finite number.
As instances of the second class, reference may be made to Euc. I. 35, 36, 37, 38, where all the parallelograms in the two former, and all the triangles in the two latter, are asserted to have the property of being equal to one given parallelogram or one given triangle. Or to Euc. III. 14, 20, 21; the lines in the circle in Prop. 14, or the angles at the circumference in Props. 20, 21, are any whatever, and therefore all the lines or angles constituted as in the enunciations, fulfil the conditions. Or again, in Book v. the two pairs of indefinite multiples, which form the basis of Euclid's definition of proportionals; or his propositions "ex æquo," and "ex æquo perturbato," "and the Propositions F, G, H, K; or, lastly, Euc. vI. 2, in which the property is (really, though not formally,) affirmed to be true when any line is drawn parallel to any one of the sides of the triangle.
The very circumstance, indeed, just noticed parenthetically, prevails so much in Euclid's enunciations, as to render it clear that it was his object as much as possible to render the conditions of the hypothesis formally definite in number; and if these remarks had no prospective reference, the circumstance would scarcely deserve notice. Still, with such prospective reference, it is necessary to insist upon the fact, that however the form of enunciation may be calculated to remove observation from it, the hypothesis itself is indefinite, or includes an indefinite number of things, which an additional condition would, as in the case of the problem, have restricted either to one thing or to a certain number of things.
Sometimes too, the theorem is enunciated in the form of a negation of possibility, as Euc. 1. 7; m. 4, 5, 6, &c. These offer no occasion for remark, except the ingenious modes of demonstration employed by Euclid. All such demonstrations must necessarily be indirect, assuming as an admitted truth the possibility of the fact denied in the enunciation.
"It may be remarked that the Ancient Geometers most probably arrived at Theorems in their attempts to solve Problems. The first Geometrical enquiries must naturally have arisen in form of questions or problems, in which some things were given, and some things required to be done and in the attempts to discover the relations between the things given and the things required, many truths would be suggested which afterwards became the subjects of separate demonstration.
Both among the Theorems and Problems, cases occur in which the hypotheses of the one, and the data or quæsita of the other, are restricted within certain limits as to magnitude and position. Sometimes it will be found, while some Problems are possible within definite limits, that certain magnitudes involved increase up to a certain value, and then begin to decrease; or decrease down to a certain value, and then begin to increase. This circumstance gives rise to the question of the greatest or least value which certain magnitudes may admit of, in indeterminate Problems and Theorems. The determination of these limits constitutes the doctrine of Maxima and Minima. For instance, the limit of possible diminution of the sum of the two sides of a triangle described upon a given base, is the magnitude of the base itself, Euc. I. 20, 22: And of all the equal triangles upon the same base and between the same parallels (Euc. 1. 37.), that triangle which has the greatest vertical angle is an isosceles triangle; and the vertical angles of the other triangles on each side of the vertex of the isosceles triangle, become greater and greater as the vertices of these triangles approach the vertex of the isosceles triangle. When a straight line is divided into two parts, the rectangle contained by the parts is a maximum when the given line is divided into two equal parts. The line AB (Euc. II. 5. fig.) is divided into any two parts in the point D. If BD the smaller part of the line be supposed to increase by the point D moving towards C, it is obvious that as the smaller part BD increases, the larger part AD diminishes, until the point D coincides with C, and both parts are then each equal to half the line AB. And it is clear that so long as BD increases, the rectangle AD, DB increases, and the square on CD decreases: and when D coincides with C, the square on CD vanishes, and the rectangle AD, DB, then becomes the square on DC, or on DB, the square on half the line AB.
If the point D be supposed to move beyond C towards A, the rectangle AD, DB begins to diminish, and the square on DC to increase, in the same manner as they increased and decreased when the point was considered to move from B to C. Hence it is manifest that when a line is divided into two parts, the rectangle contained by the parts is a maximum or the greatest possible, when the two parts of the line are equal. It also appears that when the rectangle contained by the two parts of a line is a maximum, the sum of the squares on the parts is a minimum. For if a line be divided into any two parts (Euc. II. 4.), the square on the whole line is equal to the squares on the two parts and twice the rectangle contained by the parts. Hence it follows that the greater the rectangle contained by the two parts of the line, the less will be the sum of the squares on these parts. Therefore when the rectangle contained by the parts is a maximum, the sum of the squares on the two parts is a minimum. That is, the sum of the squares on the two parts of a line is a minimum when the line is bisected; and the minimum value is double the square on half the line.
The two propositions, Euc. III. 7, 8, afford instances of the greatest and least lines which can be drawn from a given point to the circumference of a circle.
The straight line (Euc. III. 8, fig.) drawn from a fixed point without a circle to the circumference is a maximum when it passes through the center and meets the concave circumference; but a minimum when it meets the convex circumference and, if produced, would pass through the center. It is obvious, that the two values of the line on each side of the minimum value, are both greater than that value: and the two values of the line on each side of the maximum value, are both less than that value. In other words, the magnitude of the line as it approaches the minimum, continually decreases till it reaches that value, and then increases: and the value of the line as it approaches the maximum, continually increases till it reaches that value and then decreases.
When the given point is within the circle (Euc. III. 7), the greatest line that can be drawn from the point to the circumference is the line which passes through the center, and the least line that can be drawn from the same point, is the part produced of the greatest line between the given point and the circumference.
The theorem Euc. vI. 27 is a case of the maximum value which a figure fulfilling the other conditions can have; and the succeeding proposition is a problem involving this fact among the conditions as a part of the data, in truth, perfectly analogous to Euc. 1. 20, 22.
The doctrine itself was carefully cultivated by the Greek Geometers, and no solution of a Problem or demonstration of a theorem was considered to be complete, in which it was not determined, whether there existed such limitations to the possible magnitudes concerned in it, and how those limitations were to be actually determined.
Such Propositions as directly relate to Maxima and Minima, may be proposed either as Theorems or Problems. For the most part, however, it is the more general practice to propose them as Problems; but this has most probably arisen from the greater brevity of the enunciations in the form of a Problem. When proposed as a Problem, there is greater difficulty involved in the solution, as it is required to find the limits with respect to increase and decrease; and then to prove the truth of the construction: whereas in the form of a Theorem, the construction itself is given in the hypothesis.
It may be remarked that though the Differential Calculus is always effective for the determination of Maxima and Minima, (in cases where such exist) yet in many cases, where it is applied to the Problems which were cultivated by the Ancient Geometers, it is far less direct and elegant in its determinations than the Geometrical methods.
Now if reference be made to what has been stated respecting Theorems, where the hypothesis is indeterminate, or wanting in that completeness which reduces the property spoken of to a single example of the figure in question, a consequence of that peculiarity in such classes of Propositions may be remarked. This peculiarity introduces another class of Propositions, which, though in "the Elements" somewhat disguised, formed an important portion of the Ancient Geometry:the doctrine of Loci.
If the converse of Euc. 1. 34, 35, 36, 37, and Euc. ш. 20, 21, be taken in the form of Problems, they will become :
Given the base and area, to construct the parallelogram. 2. Given the base and area, to construct the triangle.