objects, the last is perhaps the most important in a course of liberal education. For this purpose, the Geometry of the Greeks is most powerfully recommended, as bearing the stamp of that acute people, and displaying the finest specimens of logical deduction. Some of its conclusions, indeed, might be reached by a sort of calculation; but such an artificial mode of procedure gives merely an apparent facility, and leaves no clear or permanent impression on the mind.

We should form a wrong estimate, however, did we consider the Elements of Euclid, with all its merits, as a finished production. That admirable work was composed at the period when Geometry was making its most rapid advances, and new prospects were opening on every side. No wonder that its structure should now appear loose and defective. In adapting it to the actual state of the science, I have therefore endeavoured carefully to retain the spirit of the original, but have sought to enlarge the basis, and to dispose the accumulated materials into a regular and more compact system. By simplifying the order of arrangement, I have materially abridged the labour of the student. The numerous additions that are incorporated in the text, so far from retarding, will rather facilitate his progress, by rendering more continuous the chain of demonstration.

The view which I have given of the nature of Proportion, in the Fifth Book, will contribute, I hope, to remove the chief difficulties attending that important subject. The Sixth Book, which exhibits the application of the Doctrine of Ratios, contains a copious selection of propositions, not only beautiful in themselves, but which pave the way to the higher branches of Geometry, or lead immediately to valuable practical results. The Appendix, without claiming the same degree of utility, will not perhaps be deemed the least interesting portion of the volume, since the ingenious resources which it discloses for the construction of certain problems are calculated to afford a very pleasing and instructive exercise.

The Elements of Trigonometry are as ample as my plan would allow. I have explained fully the properties of the lines about the circle, and the calculation of the trigonometrical tables ; nor have I omitted any proposition which has a distinct reference to practice. Some of the problems annexed to the treatise are of essential consequence in marine surveying.

In the improvement of this edition, I have spared no trouble or expence. The text has been simplified and reduced to a shorter compass, by throwing such propositions as were less elementary into the Notes. Other Notes of an easier kind are intended chiefly to engage the attention

of the young student. In various parts of the work, the demonstrations are occasionally abbreviated. The Elements of Trigonometry are now much expanded, and brought to include whatever appears to be most valuable in recent practice. But the principal additions have been made in the Notes and Illustrations, which will be found to contain a great variety of useful and curious information. The more advanced student may peruse with advantage the historical and critical remarks; and some of the disquisitions, with the solutions of certain more difficult problems relative to trigonometry and geodesiacal operations, in which the modern analysis is but sparingly introduced, are of a nature sufficiently interesting to claim the notice of proficients in science. I have simplified, and materially enlarged the formulæ connected with trigonometrical computation; explained the art of surveying, in its different branches; and given reduced plans, blended with the narrative of the great operations lately carried on both in England and France. I have likewise shown a very simple method of calculating heights from barometrical observations, accompanied by illustrative sections; and I have been thence led to state the law of climate, as it is modified by elevation. On this attractive subject, I should have dwelt with pleasure, had the limits of the volume permitted.

To trace the silent progress of discovery, is at once interesting and instructive. I have therefore laboured to set in a clear light the Trigonometry of the Greeks and Arabians, and have carefully marked the successive steps by which this important branch of science was, in passing from ancient to modern times, advanced to perfection. In these critical inquiries, I have derived essential aid from the extensive and accurate researches of M. Delambre, whose learning, patience, and discernment, are above all praise. It has afforded me the highest satisfaction in finding the opinions I was led to form on several disputed points of scientific history generally to accord with the mature conclusions of that eminent philosopher.

In explaining the division of the circle, I have introduced some short tables, which will furnish an useful exercise to the student; and the examples I have given of the conversion of sexagesimals into decimals will show how much greater nicety the ancient Greeks had attained in their calculations than is commonly supposed. I have still farther enlarged the trigonometrical formulæ, and have applied the tables of sines and tangents to the solution of quadratics and the irreducible case of cubic equations. The successive attempts made, at different periods, to measure the extent of our globe are now distinctly related, and the most improved methods of conducting such geo

graphical surveys are explained and exemplified, I presume, with sufficient detail. Other additions, either curious or illustrative, will occur in various parts of the work.

My original design was to exhibit, within the compass perhaps of five volumes, the Elements of Mathematical Science in their full extent, including the principles and application of the Higher Calculus. But, after due reflection, I have abandoned that aspiring project. There is unfortunately very little incitement to the publication of abstract works in this country; and after discharging the more pressing obligations which I had contracted, I shall consider my time as more agreeably and perhaps more beneficially employed, in pursuing without distraction the labyrinths of physical research. I might have foreseen that the indolence of teachers would always be opposed to the improvement of education; yet I have very lately revised, and somewhat enlarged, the Philosophy of Arithmetic, which I am convinced will form the most instructive introduction to Algebra and to the science of calculation in general. In a few weeks another volume will be delivered from the press, containing the tract on Geometrical Analysis, recast and augmented; the Geometry of Curve Lines, including not only a regular and complete system of the Conic Sections, but exhibiting the beautiful relations of those Higher Curves, ancient or modern, which either invite the application of