century, geometrical questions of this kind seem to have been proposed: there is one, in particular, sent from Fermat to Torricelli, the solution of which will be given in the course of this work. Dr. Barrow, in the Addenda to his Lectiones Geometricæ, a work of great originality and erudition, has shewn how theorems relating to maxima and minima may be deduced from the consideration of tangents. His two fundamental theorems are demonstrated with admirable perspicuity and elegance. Amongst the works of later mathematicians, not to mention those of L'Huillier and other foreigners, the Elements of Thomas Simpson contain a series of propositions, "on the Maxima and Minima of Geometrical Quantities," in which there is not much that is original. Waring, in his Treatise on Curve Lines, has determined several maxima and minima, of which it may be said, that they are curious rather than useful. But the two last-mentioned authors, as well as Pappus among the ancients, and most of the modern writers upon this subject, have admitted into their reasonings what appears to be a sophism. It is easily shewn, for example, that of all triangles standing upon the same base and of equal perimeter, that which is isosceles is the greatest: hence they have concluded, at once, that whatever the rectilineal figure be, of a given perimeter, when it is greatest, its sides are all equal. Thus they have taken for granted that the quantity under consideration has a maximum value: which is not allowable in any geometrical proposition. Such an assumption may be properly made in the analytical investigation of maxima and minima; because if no maximum nor minimum exists, the process itself will shew that to be the case, by the nature of its result: and if no absurdity appear in the result, and it still be doubtful whether the quantity determined give a maximum or a minimum value, there are means of ascertaining to which of the two it belongs. But, in this application of Geometry, it is necessary to prove that the variable magnitude is greater, or less, than any other of the same kind with itself, before the conclusion can be fairly drawn. The first division of the following publication is purely geometrical, and an easy application for the most part of the Elements of Euclid. Wherever any theorem or problem is wanted, which is not contained in that book, it has been supplied: in those cases, in which a proposition relating to maxima and minima appears to depend principally upon some more simple geometrical truth, this latter has been separately premised; in order that it may be distinctly seen upon what elements each main proposition is founded. From a wish to accommodate this work, as far as it could well be done, to those who have studied only the first four books of Euclid, the doctrine of proportion has been, as much as possible, avoided: although the use of it might have shortened some of the demonstrations. The propositions of the first and second Sections, of this first Part, form a distinct and important subject: they lead to results which have, most of them, been long known, but which are, perhaps, no where to be found collected, arranged, and strictly demonstrated. The maxima of the first Section are, each of them, connected with a minimum: that is, the same species of figure. which renders the surface greatest when the perimeter is given, renders the perimeter least when the surface is given. This remarkable property is shewn, in a general theorem, necessarily to obtain. In the questions of the second Section, on the contrary, the area is a maximum when the perimeter is a maximum; and it is a minimum when the perimeter is a minimum. In one description of them, whilst the perimeter remains the same in length, the area also remains the same, whatever be the number of sides of the figure. The third Section consists of miscellaneous propositions; classed, however, according to division, which refers them to lines, angles, and surfaces. It could not escape observation, if the mention of it were suppressed here, that it is part of the plan of this work to invite a comparison between Geometry and Algebra, and to illustrate the advantages peculiar to each. The relative advantages of these two great branches of science, in the investigation of mathematical truths, are now, indeed, well understood. But it may not be improper to offer some remarks on the great difference which there is between them in producing those collateral effects, which have been ascribed to the mathematics, considered as a discipline of the mind. In the very entrance upon our discussion of this topic, in order to avoid all misconstruction, it may not be wholly needless to state expressly, that what follows is intended to refer solely to the case of academical students, who apply themselves to the mathematics, not so much on account of the intrinsic value of that science itself, as for the sake of those indirect advantages, which are supposed to flow from the cultivation of it; such as the habits of close attention, of weighing the validity of proofs, of searching into the connexion of related truths, and of methodizing the materials of thought. With this particular view, then, let it be remembered, our enquiry is to be conducted and it will turn principally on the comparative merits of the analytic and the synthetic modes of reasoning, so considered. Both these modes of reasoning may, indeed, be used in every department of the mathematics. But throughout the whole province of arithmetic, numerical as well as algebraical, elementary as well as infinitesimal, both in the investigation of theorems and in the solution of problems, the analytic method is, almost exclusively, employed whilst the truths of Geometry are, for the most part, demonstrated synthetically; and the student in acquiring them becomes habituated to the use of that method of teaching. Now, there exists, in the first place, this manifest distinction between a synthetic proof in Geometry, and an analytic process in Algebra, that in order to comprehend the former, the whole chain of reasoning must be kept in view, as it is continued from the beginning of the proposition to the end whilst in pursuing the latter method, the attention is fixt only upon each single step, as each of them successively offers itself; and the conclusion is to be admitted independently of all but the last of them, whenever it is arrived at. Stronger and more unceasing attention, therefore, is required in the former case, than in the latter, and the judgment, as well as the memory, is called more urgently into action. There is, however, analysis, as well as synthesis, in Geometry. All those propositions, the truth of which Euclid has deduced ex absurdo, are, in reality, demonstrated analytically: and, in the same manner, a series of conclusions legitimately drawn from a certain supposition, may so terminate as to shew that supposition to be true. But, in both these cases, it is evident that the connexion of the several steps with the original hypothesis must be closely attended to, in order that the force of the proof may be clearly seen. Arithmetical speculations, on the contrary, most commonly hinge upon the solution of an equation, or upon the finding of a fluent and whatever obstacles the authour of such propositions may have had to encounter, there is seldom any serious difficulty in following him along the path which he has traced out. |