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exercise the pupil in the various branches of mathematics: comprised in the course, to demonstrate their utility especially to those devoted to the military profession, to excite a thirst: for knowledge, and in several important respects to gratify it.. This volume being professedly supplementary to the pres ceding two volumes of the Course, may best be used in tuition by a kind of mutual incorporation of its contents with those: of the second volumes. The method of effecting this will, of course, vary according to circumstances, and the precise? employments for which the pupils are destined: but ingeneral it is presumed the following may be advantageously adopted. Let the first seven chapters be taught immediately after the Conic Sections in the 2d volume. Then let the substance of the 2d volume succeed, as far as the Practical Exercises on Natural Philosophy, inclusive. Let the 8th and 9th chapters in this 3d vola precede the treatise on Fluxions s in the 2d; and when the pupil has been taught the part res lating to fluents in that treatise, let him immediately be conducted through the 10th chapter of the 3d volume. After he has gone over the remainder of the Fluxions with the applications to tangents, radii of curvature, rectifications, quadratures, &c, the 11th and 12th chapters of the 3d vol. should be taught. The problems in the 13th and 14ths chapters must be blended with the practical exercises at the end of the 2d volume, in such manner as shall be found best suited to the capacity of the student, and best calculated to ensure his thorough comprehension of the several curious problems contained in those portions of the work.
In the composition of this 3d volume, as well as in that of the preceding parts of the Course, the great object kept constantly in view has been utility, especially to gentlemen intended for the Military Profession. To this end, all such investigations as might serve merely to display ingenuity or talent, without any regard to practical benefit, have been care
fully excluded. The student has put into his hands the two powerful instruments of the ancient and the modern or sublime geometry; he is taught the use of both, and their relative advantages are so exhibited as to guard him, it is hoped, from any undue and exclusive preference for either. Much novelty of matter is not to be expected in a work like this; though, considering its magnitude, and the frequency with which several of the subjects have been discussed, a candid reader will not, perhaps, be entirely disappointed in this respect. Perspicuity and condensation have been uniformly aimed at through the performance: and a small clear type, with a full page, have been chosen for the introduction of a large quantity of matter.
A candid public will accept as an apology for any slight disorder or irregularity that may appear in the composition and arrangement of this Course, the circumstance of the different volumes having been prepared at widely distant times, and with gradually expanding views. But, on the whole, I trust it will be found that, with the assistance of my friend and coadjutor in this supplementary volume, I have now produced a Course of Mathematics, in which a greater variety of useful subjects are introduced, and treated with perspicuity and correctness, than in any three volumes of equal size in any language.
CONTINUATION OF THE CONIC SECTIONS.
IN the year 1787 was published, by order of the Master
General of the Ordnance, for the use of the Royal Military Academy, a volume of miscellaneous exercises, which had, for many preceding years, been employed in manuscript, in the education of the cadets in the academy. The first and principal article in the contents of that volume, was an extensive geometrical treatise on Conic Sections, treated in a new and a more methodical, as well as easier way, than had been usual. In the year 1798, when the 2d volume of the Academical Course was first published, by order of the Master General also, the leading propositions of that treatise on Conic Sections were introduced into it.-And now, on the further extension of the Course, by order of his lordship the present Master General the remaining propositions, of the said first treatise of Conics, are introduced into this 3d volume.
It will be observed that the theorems or propositions in this volume, are numbered in the regular succession from those in the 2d volume, in each of the three sections, commencing here, in the 3d volume, with the number next following the last in the 2d volume, so as to form these propositions in both VOL. III.