results; the modern geometry is a collection of general truths, each comprising under it an endless number of particulars.

We have spoken of geometry as the science which has for its object the measure of extension. This definition, though it may seem at first sight, by its precision, to limit the scope of geometry, does in reality require, for the absolute perfection of this science, that it should discuss all imaginable forms of lines, surfaces and volumes, and discover all the properties which belong to each form.* This statement immediately suggests two essentially distinct modes of investigation; the one by taking up, one by one, these geometrical forms, and determining separately all the properties of each; the other, by grouping together the discussion of analogous properties, no matter how different in other respects may be the bodiest to which they belong. In other words, our geometrical researches may be conducted, and the results of them arranged in relation to the different bodies which are the object of study, or in relation to the properties which these bodies present. The first of these was the method pursued by the ancients. They studied, one by one, the properties of the straight line, the circle, the ellipse, the hyperbola, &c., separating the different questions pertaining to each from those which related to other curves or surfaces, no matter how strong the analogies might be between them. This method of investigation, though simple and natural, is obviously characteristic of the infancy of science. The complete mastery of the properties of one curve affords no aid for discovering those of another, beyond the skill and tact which the previous study has imparted. No matter how similar may be the questions discussed respecting different curves, the complete solution of them in relation to one leaves us to commence investigation anew for every other. However similar a problem may be to one already solved for some other curve, we can never be certain beforehand that we shall have sufficient address to solve it under its modified form. Though we may, for example, have learned how to draw a tangent to an ellipse or hyperbola, this gives us no aid in determining the tangent to any other curve. Geometry, thus studied, is, as we have already called it, evidently nothing more than a collection of particular results, destitute of those general classifying truths which are necessary to constitute a science.

The modern geometry, on the other hand, instead of investigat ing seriatim the properties of each geometrical form, groups together all affections of a like kind and discusses them without regard to the particular bodies to which they belong. It passes over, for instance, the particular problem of finding the area of the circle, and solves the general problem of finding the area bounded by any

For a lucid exposition of this and some other points briefly discussed in this article, the reader is referred to M. Comte's Cours de Philosophie Positive, Leçon

10e.

We use the term body, for convenience sake, to designate the objects of geometrical study, lines, surfaces and volumes.

curve line whatever. Instead of investigating the asymptote to the hyperbola, and then remaining in no better condition than before for discovering whether any new curve has asymptotes or not, it puts us in possession at once of a general method for determining the asymptotic lines, straight or curved, which belong to any curve whatever. The modern geometry treats thus, in a manner perfectly general, every question relative to the same geometrical property or affection, without regard to the particular body to which it may belong. The application of the general theorems thus constructed, to the particular circumstances of this or that curve or surface, is a work of subordinate importance, to be executed according to certain rules that are invariable in their mode of application and infallible in their promise of success.

Let any new curve be proposed to one who is destitute of the resources of the modern geometry, and he must commence first by surmising, and that chiefly through the suggestive power of graphical constructions, what its properties are, and then endeavour to prove by methods altogether peculiar to the curve in hand, that it possesses the properties the existence of which he has divined, with no certainty derived from his previous knowledge that he will be able to succeed in this particular case. Foiled amid its intricate specialities he may be reduced, as was the great Galileo, to the mortifying necessity of calling in the mechanical aid of the scales to supply the defect of his mathematical resources.* Let the same curve be proposed to one who has the modern geometry at command, and he will immediately determine its tangent, its singular points, its asymptotes, its radius of curvature, its involute and evolute, its caustics, its maximum and minimum ordinates, its length, its area, the content of the solid generated by its revolution, in short all its important properties.

The brief exposition which we have given of the different methods pursued by the ancient and the modern geometry, is enough to show on which side the scientific superiority lies. In the ancient geometry special results are obtained separately, and without any knowledge of their mutual relations, though they may be, in truth, only particular modifications of some general truth which embraces them and innumerable like phenomena. The modern geometry investigates this general truth, and then applies it, in the way of deduction, to all particular cases. Had we gone on for ages in the steps of the ancients, we could have done nothing more than add to the indigesta moles of particular truths; and no matter how great our success there would still always remain an infinite variety of geometrical forms unstudied and unknown. On

*The only stain upon the scientific reputation of this great man is his seeking to determine the area of the cycloid in terms of its generating circle, by cutting the cycloid and the circle out of a lamina of uniform thickness and weighing them. It is a striking illustration of the power of the modern analysis that any tyro can now solve problems that eluded the forces of such men as Galileo, Fermat, Roberval, and Pascal.

the other hand, for every question resolved by the modern geometry, the number of geometrical problems to be solved is diminished, for all possible bodies. The one is a science, with its general theorems lying ready for all possible cases; the other is made up of independent researches, which, when they have gained their particular end, shed no light beyond it.

It is not our purpose to enter fully into the exposition of the peculiar logic of the modern analysis, or to contrast in detail its merits with those of the ancient geometry. Many interesting points of view could be obtained by pursuing this comparison to a greater length; but we have gained the end which we at present have in view if we have given an exposition of the subject sufficiently plain and extended to enable the reader to pronounce upon the scientific claims of the two methods. We entertain no doubt what will be the judgment rendered.

The superiority of the analytical methods of the moderns is so evident and vast, that there has been no attempt, since the publication of the "Geometry of Curve Lines," by Professor Leslie, to revive the ancient method. This attempt was a signal failure. Mr. Leslie avows himself the champion of a juster taste in the cultivation of mathematical sciences, but unfortunately for his success, no sooner does he enter upon any question which lies beyond the mere elements of geometry than he betrays most painfully the poverty of his resources. We have but to open his book and read of "a tangent and point merging the same contact," of points "absorbing one another," of "tangents melting into the curve," of "curves migrating into one another," &c., to make us sympathize with the humiliation which he must have felt in invoking the aid of poetry to establish the theorems of geometry. We know of no similar attempt made by any scholar since. It is now universally conceded that without the aid of the modern analysis, the science of geometry cannot be established upon a rational basis. And without the help of geometry, thus established and ordered, all the real sciences, excepting only those included in the department of natural history, must be deprived of their full development and perfection. The new geometry has its ample vindication in the" Mécanique Analytique" of Lagrange, and the "Mécanique Céleste" of Laplace.

In our own country, prior to the publication of the work named at the head of this article, we had but two treatises on the subject of Analytical Geometry; the one a republication of the elementary treatise of Mr. J. R. Young, which is chiefly made up from the "Application de l'Algèbre a la Géométrie" of Bourdon; the other, a more recent publication from the pen of Prof. Davies. We do not, for reasons that will be obvious enough, include among treatises upon Analytical Geometry, the Cambridge translation of the imperfect and antiquated work of Bèzout. We are glad that Prof. Smith has added his contribution to our scanty stock, by giving us a translation of the masterly work of Biot, one of the

most perfect scientific gems to be found in any language. The original needs not our commendation, and of the translation it is enough to say that it is faithfully executed.*

We regard the multiplication of text books, on this subject, as affording cheering evidence that juster ideas are beginning to prevail in our country respecting the proper scope of mathematical education. And yet there are colleges in our land that comprise, in their course of study, nothing of the geometry of curves beyond what is contained in Simpson's or Bridge's Conic Sections, that leave the study of the Calculus optional with the student, and that are compelled, therefore, to teach, under the name of Natural Philosophy, a system that, at the present day, is scarcely level with the demands of a young ladies' boarding school. The graduates of these institutions may be able to classify plants, insects, and stones; they may fancy themselves qualified to decide upon the comparative merits of rival systems of world-building in geology; but they cannot read, understandingly, the first ten pages of any reputable treatise on mechanics from the French or English press. We have grieved long over this state of things, and we hail with pleasure every symptom of a change for the better in public sentiment. If our ancient and venerable institutions of learning will not elevate their course of study into some approximation to the existing state of mathematical science, the day, we hope, is not far distant when the public will discern that they are standing in the way of a thorough education, and visit them accordingly.

We regret to see so many typographical errors in the work, and some of them of a character fitted to perplex the student. On page 88 there is an omission of the transformation of an equation of the Ellipse, to remove the origin from the vertex of the axis to the centre of the curve, which confuses all the subsequent investigation.

ESSAY XVI.

BAPTIST TRANSLATION OF THE BIBLE.*

WHILE the existence of different religious sects in the world opens a wide field for the exercise of Christian charity, the most rational foundation for that charity is laid in the principles of the separation. Each Protestant sect admits, and with great propriety, that a way to heaven may lie through the territories of all other Christian denominations, and that every one of the numerous forms in which the truth is held and preached, may be instrumental in producing and sustaining a saving faith in Christ. We expect to find true piety in every division and under every name of the Christian church. The various denominations of Christians, which have gained any considerable note in the world, have kept up by means of their forms of worship, doctrine and order, their broad distinctions from one another; while, as to degrees of practical piety, no one of these prominent and prosperous sects has probably varied more from the others, than the same sect has, in different times and circumstances, varied from itself. We are, therefore, as reasonably bound to cultivate a fervent charity towards the members of other denominations as towards those of our own. We know not at what point in the progress of the sincere but mistaken upholders of error, our charitable regards should stop. In this state of mingled truth and error, it is impossible for man to fix the precise line where the light of saving truth is bounded by the verge of total darkness. No mere man since the fall can be sup

Originally published in 1838 in review of, 1. "Constitution of the American and Foreign Bible Society, formed by a Convention of Baptist Elders and Brethren, held in the Meeting House of the Oliver street Baptist Church, New York, May 12 and 13, 1836.

2. " Proceedings of the Bible Convention of Baptists held in Philadelphia, April 27-29, 1837.

3. "Report of the Board of Managers of the American and Foreign Bible Society, embracing the period of its Provisional Organization. April, 1837.

4. "Christian Review and Translations of the Bible, Nos. 5 and 8. March and December, 1837.

5. "First Annual Report of the Executive Committee of the American and Foreign Bible Society, presented April, 1838.”