cating it in a pleasing popular manner. Indeed he is almost the only British mathematician since the times of Barrow, Maclaurin, Halley, and Cotes, who has not suffered his love of abstract investigations to weaken his attachment to the graces. His various performances already before the public, (inore especially his admirable Introduction to Astronomy, and the preface to his Algebra) demonstrate that he is alike versant in the intricacies of science, the embellishments of poetic genius, and the fascinations of a glowing yet classical style.

The work now before us appears to be the result of much labour and investigation; and though employed on a subject that scarcely admits of decoration, it will be found to exhibit strong traces of the hand of a master. It is preceded by an introduction of 28 pages, containing a concise, but judi-. cious and interesting history of the principal writings relative to logarithms and trigonometry. The next seventy pages exhibit the most important definitions, and the logrithmic and algebraic rules for all the cases of right and ob lique trigonometry; with a great variety of practical examples, several of which are wrought out at length. These are followed by the, doctrine of spherical trigonometry, and its application to astronomical problems, comprized in about two hundred pages. Mr. Bonny castle then treats the mutations of the signs of trigonometrical quantities; and afterwards furnishes, to the extent of 50 pages, a very copious and valuable collection of trigonometrical formulæ, relating to what is frequently called the arithmetic of sines and cosines, the values of sines, cosines, &c. in terms of circular arcs, &c. and vicè versâ, expressions for the verification of tables of sines and tangents, exponential quantities, logarithmic series, series for multiple arcs, and for logarithmic sines, tangents, secants, &c.

The trigonometrical and logarithmic formulæ are followed by demonstrations of the principal theorems in plane and spherical trigonometry, exhibited and exemplified in the earlier part of the volume; and these are followed by de monstrations of the leading theorems in the stereographic projection of the sphere, some miscellaneous problems relative to spherical areas and the deduction of what is known by the name of the spherical excess,-solutions of all the cases of plane triangles, independently of any tables, by means of series and approximating expressions,-formulæ shewing the relations of the fluxions and increments of the sines and tangents of arcs and angles,-the solutions of quadratic, and cubic equations by tables of sines and tangents, and rules (we believe, those of the late Professor Robison) for the ad

measurement of altitudes by the barometer and thermo

meter.

This volume is marked with several peculiarities, most of which are objects of commendation. Thus, the simplicity of the figures illustrating the definitions, the algebraic expressions for the areas of plane triangles, the perspicuity of the general rules pointing out the nature and affections of spherical triangles, and, above all, the truly admirable collection of trigonometrical formula, cannot fail, the former to be beneficial to the novice, and the latter to be highly gratifying to all who have imbibed the genuine spirit of mathematical elegance and taste. We cannot but admire the care with which Mr. Bonny castle has adapted his precepts to the present state of mathematics in this country, and his dexterity in striking into the just medium between the tiresome and uncouth prolixity of many writers of the seventeenth century, and the purely analytical process of Gua, La Grange, aud other modern mathematicians among the French. The latter indeed has numerous advantages, which, it is manifest, our author knows how to appreciate, though he has wisely chosen rather to allure the student by setting before him occasional specimens of the method, than to deter him by adopting it exclusively, to the neglect of that preliminary knowledge, which, as an experienced tutor, he well knows where to place. Besides, a very material benefit arises from exhibiting the algebraic formula for spherical triangles, as Mr. B. has done, together with the logarithmic rules; because, by duly attending to the signs (+ and --) of the various expressions for the sines, tangents, &c. of arcs or of angles, and particularly by adopting those formula which furnish results in cosines, or cotangents, or the tangents of half arcs, or tangents of 45° + half arcs, or angles, ambiguity which would otherwise arise on the resolution of spherical triangles may be avoided, except those which appertain to the two cases that are necessarily ambiguous; and even they may be guarded against by means of Bertrand's table, (referred to in a note p. 79); a table which we are sorry our author omitted, as it would not have occupied a page.

every

We have been so well satisfied with the execution of this volume, that we cannot help regretting Mr. B. did not adopt a plan somewhat more extensive; a plan which would have enabled him to treat more at large the different kinds of projection, the application of the fluxionary calculus to those trigonometrical quantities which are employed in astronomical problems, and their maxima and minima, and some other topics that are confined in too narrow a space toward the end of the present

Treatise. This however could not have been accomplished, without giving an additional volume; which might probably have circumscribed the sale, and therefore the usefulness of the work.

With a disposition perfectly friendly toward this able writer, we beg to remark, that his Treatise, while it displays very frequently such marks of taste, talents, and analytic skill, blended with a prevailing love of simplicity, as are not to be found in any other English work on the subject, betrays, notwithstanding, some tokens of carelessness. In the arrangement, we think Mr. B. would have done well to follow the plan pursued in his useful little book on Mensuration, giving the demonstrations of the rules and theorems, at the foot of the respective pages on which they are found. We should likewise have been glad if the student's perusal of this work had been facilitated by the customary help of running titles, a table of contents, or a copious alphabetical index.

Instead of dwelling on a few slight imperfections, which we are persuaded the author will remove in a future edition of his Treatise, we shail lay before the reader some inferences (in our estimation important) which we were inclined to make, in perusing the latter part of this interesting performance.

Among the trigonometrical formulæ are several of those curious and useful ones which exhibit the sine, cosine, &c. of an arc in exponential expressions deduced from the product of imaginary factors: such, for example, as

Now, while we admit the facility and neatness with which transformations may be made, and the values of linear-angular quantities may be inferred, by means of such expresssions; we would ask what definite ideas can accompany them? Can we obtain any clear comprehension, or indeed any notion at all, of the value of a power whose exponent is an acknowledged imaginary quantity, as -1? Can we, in like manner, obtain any distinct idea of a series constituted of an infinite number of terms? In each case, we are convinced, the answer will be in the negative. Yet the science in which these and numerous other incomprehensibles occur, is called Mathesis, THE DISCIPLINE, from its incomparable superiority to other studies in evidence and certainty, and, therefore, its singular adaptation to discipline the mind. And this is the science, says the eloquent and profound Dr. Barrow, "which effectually exercises, not vainly deludes, nor vexatiously torments studious minds, with obscure subtilties, perplexed difficulties, or contentious disquisitions; which overcomes

without opposizion, triumphs without pomp, compels without force, and rules absolutely without any loss of liberty; which does not privately over-reach a weak faith, but openly assaults an armed reason, obtains a total victory, and puts on inevitable chains." How does it happen, now, that when the investigation is bent toward objects which cannot be comprehended, the mind arrives at that in which it acquiesces as certainty, and rests satisfied? It is not, certainly, because we have a distinct perception of the nature of the objects of the inquiry (for that is precluded by the supposition, and indeed by the preceding statement); but because we have such a distinct perception of the relation those objects bear one toward another, and can assign positively, without danger of error, the exact relation as to identity or diversity of the quantities before us, at every step of the process. Mathematics is not the science which enables us to ascertain the nature of things in themselves;-for that alas! is not a science which can be tearnt in our present imperfect condition, where we see through a glass darkly;" but the science of quantity, as measurable, that is, as comparable and it is

66

-

obvious that we can compare quantities satisfactorily in some respects, while we know nothing of them in others. Thus, we can demonstrate that any two sides of a plane triangle are together greater than the third, by shewing that angles of whose absolute magnitude we know nothing are one greater than the other, and then inferring the truth of the proposition from the previously demonstrated proposition, that the greater angle in a triangle is subtended by the greater side. Again, we cannot possibly know ALL the terms of the infinite series,

a

a

because such knowledge implies a contradiction: neither can we know all the terms of the infinite series,

1

2

3

C

yet we can shew that these series are equal.

.c&ܕ

For we can

demonstrate that the first series is an expanded function standing with the quantity in the relation of equality:

1

a + c

we can likewise demonstrate that the second series bears the relation of equality with the quantity : and although we

1

c + a

can have, but a vague idea of the quantities

1

c + a

while a and c stand as general representatives of any quan

tities; yet those fractions must necessarily be equal, and thence we infer the like equality between the sums of the two infinite series. In the same manner, if we revert to that kind of quantities which we first noticed, we can have no clear conception of the nature of the quantities -a,

√—b, &c. yet we are as certain that a = √ − b x √ — √-a=√-6X

as that 20+30=50; since we can demonstrate that equality subsists in the former expression as completely as we can in the latter, both being referable to an intuitive truth. Every mathematician can demonstrate strictly that the conclusions he obtains by means of these quantities, though he cannot comprehend them in themselves, must necessarily be true: he therefore acts wisely when he uses them, since they facilitate his inquiries; and knowing that their relations are real, he is satisfied, since it is only in those relations that he is interested.

Our great object in this induction of particulars is, to recommend that similar principles be adopted when religious topics are under investigation. We cannot comprehend the nature of an infinite series, but we know the relation which subsists between it and the radix from which it is expanded; we cannot comprehend the nature of the impossible quantities √—a, √—b, &c. but. we know their relation to one another and to other mathematical quantities. In like manner, (though we should scarcely presume to state such a comparison, but for the important practical inference which it seems to furnish) we cannot with our limited faculties comprehend the infinite perfections of the Supreme Being, or reconcile his different attributes, so as to see distinctly how mercy and peace are met together, righteousness and truth have embraced each other," or how the Majestic Governor of the universe can be every where present, yet not exclude other beings; but we know or may know his relation to us, as our Father, our Guide, and our Judge ;-We cannot comprehend the nature of the Messiah, as revealed to us in his twofold character of "the Son of God" and the "Man Christ Jesus;" but we know the relation in which he stands to us as the Mediator of the New Covenant, and as he "who was wounded for our transgressions, who was bruised for our iniquities, and by whose stripes we are healed:"Again, we cannot comprehend why the introduction of moral evil should be permitted by him "who hateth iniquity;" but we know, in relation to ourselves, that he hath provided a way for our escape from the punishment due to sin, and therefore, though we cannot comprehend and explain it so as to silence all cavillers, yet we "glory in the mystery of Recon