to think in another language, and to arrive in a moment at results, that can only be reached by any only be reached by any other process after years of painful investigation." The truth of the Calculus none now doubts, but who can comprehend some of its teachings? An infinite area, enclosed between a straight line and a curve, whose equation is y2=1, is shown by the method of quadratures to be equal to a square whose surface is 2. The finite equal to the infinite! Does the mathematician reject the conclusion because he cannot understand it? Not at all. Does he throw away his Treatise on Fluxions and pronounce the whole false? Not at all. How then, can he, with any sort of propriety, reject the Bible, because he cannot understand the doctrine of the Trinity, or the Divinity of the lowly Nazarene? Again. Newton regarded all lines, whether straight or curved, as having been generated by the motion of a flowing point. Thus, to illustrate his meaning, a point, moving or flowing with the condition of being in the same plane and always at the same distance from another point, will generate the circumference of a circle. He also considered lines, straight and curved, as the generatrices of surfaces, and surfaces, in their turn, the generatrices of solids. The point, then, is the source of all lines, surfaces and solids; the flowing point generates all geometrical bodies. This was Newton's theory, and he made it the basis of the Doctrines of Fluxions and Fluents, so called from this very fact. Now, a geometrical point is an ideal thing, "it has neither length nor breadth,, but position only." So, then, an immaterial thing is the ultimate source of all geometrical magnitudes! When Euclid was asked by Ptolemy, King of Egypt, if the science of geometry might not be made easier, he replied, "there is no royal road to geometry." There was no royal road then, but more than fifteen hundred years after Euclid, Francis Vieta found in Algebraic Analysis a royal road to Geometry which even the weakest king may walk in. He showed that the most difficult problems of Geometry could be solved by a few simple operations upon an equation. Analytical Geometry became a science, and has claimed the admi ration of the greatest mathematicians for more than five centuries. But do not some of its conclusions appear absurd? Parallel lines, and none but parallel lines, are shown to meet in infinity; the assymptote is not parallel to, yet it meets the hyperbola in infinity. The Hyperbolic spiral is generated by a point, which starts in infinity and eternally approaches toward another point, every moment becoming nearer and yet never reaching it. The shorter the supplemental chord of the eclipse is made, the greater becomes the angle between it and the transverse axis, and finally when the chord is reduced to a point, the angle becomes equal to ninety degrees. Then the angle between a point and a straight line is a right angle! How absurd! But who so silly as to abandon Analytical Geometry, because of its incomprehensible truths? At every step in mechanics, astronomy, and the higher mathematics, the scientific scholar meets with results which appear absurd, irreconcilable and impossible. Still his faith in science is not shaken, his devotion is not weakened. How is it possible then, for him to reject the Bible, because of its mysteries? Tis nonsense to suppose it. True science never yet made one sceptic. Love of sin has made thousands. If Laplace and a few other mathematicians were infidels, 'twas not because of their scientific attainments, but because their corrupt hearts hated the Book that taught, "the soul that sinneth, it shall die." 3d. Mathematical studies sober the fancy, and tend to repress wild, foolish and extravagant speculations. Next to the aversion of the corrupt heart to holiness and truth, the greatest enemy with which Christianity has had to contend, since its first promulgation by the Son of God, has undoubtedly been the speculations of Philosophers, "falsely so called." Speculation has slain its thousands, and love of sin its tens of thousands. The philosopher first makes his theory about the age of the world, the unity of the human race, the moral right of involuntary servitude, etc., and then examines his Bible, to see whether its infallible teachings accord with his silly vagaries. Finding nothing in its holy pages to pamper, and everything to rebuke, the wild riotings of his imagination, he rejects God's inestimable word, raVOL. VIII.-No. 1. 3 'that men, who are accustomed to receive only demonstrated truths, are not liable to be misled by a fanciful theory, and that men, whose pursuits call for the exercise of the reasoning faculties only, are not apt to give the reins to imagination. Thus mathematicians are rarely capable of producing, or even of relishing poetry. Since the world was made, there never was an epic produced by one of the class, and Playfair, we believe, was the only man of science for three centuries, who could write a sonnet to a lady's eyes. Now, if it be true that the imagination of the mathematician is in abeyance to the reasoning faculties, and if an excess of imagination be unfavorable to religion, it follows that mathematical studies tend to promote rational piety. None will question that mathematicians have but little imagination. We then only have to examine the effects of letting loose the reins of the fancy. Iræneus tells us in so many words, that the great Gnostic heresy, in the primitive ages of the church, was the natural fruit of letting the imagination run riot, and that their doctrine of eons, spiritual emanations, was the offspring of a diseased fancy. Bishop Watson, in his reply to Gibbon, says, in speaking of modern infidels, "they are all miserable copiers of their brethren of antiquity; and neither Morgan, nor Tindal, nor Bolingbroke, nor Voltaire have been able to produce a single new objection not advanced by the Gnostics." Now, if Iræneus and the Bishop are right, the unchecked fancy has been the source of most of the heresy and scepticism in the world. They are high authority, and here we might rest the matter, but a few more facts will strengthen our position. Hume, says his biographer, was led by his fondness for speculation, and his love of applause, to attempt the subversion of all that the Christian holds sacred. He doubted everything, and then doubted whether he doubted. And thus he floundered in the meshes of speculative philosophy, till death and the realities of eternity solved all his doubts. Bishop Berkely wrote an essay against mathematics, and a treatise on mental sciAnd what has the world gained by the teachings of the hater of mathematics? Why, it has been taught ence. that all is spirit, and that there is no matter in the uni verse. "When Bishop Berkely said 'there was no matter,' But there is this much matter in his foolery. It shows what a dangerous thing an unrestrained imagination is, when it can make even a learned prelate rave like a maniac. Lord Monboddo was deficient in mathematical attainments, but eminent as a Greek scholar and metaphysician. He believed that man was but an elevated species of monkey, an improved edition of the ourang outang. Had the theorist been a better mathematician, he would have known that the converse of a proposition is often true, when the direct proposition is false. That though monkeys never rub off their tails and attain to the dignity of the lords of creation, yet the ingenious speculator, and thousands of others like him, have put on the airs and grimaces of the ape, and degraded themselves to the level of the baboon. Other theorists, equally as wise as Lord Monboddo, have supposed that a man was a compound of the whole animal creation, and that the predominance of any one animal determined the character of the individual. Thus the mean man has the dog in excess, the rude man too much bear, the slovenly man too much hog, &c., &c.; thus too, old maids have invariably, either the lamb or the viper, out of all proportion to other ingredients in their composi tion. Such have been the follies, into which the greatest intellects of the world have fallen, when they let their fancy go unchecked. Ought we not to be thankful that there is a class of men, in whom, reason always sways the imagination? Are they not, of all men in the world, the least liable to fall into scepticism, and most apt to relish the sober teachings of the word of God? 4th. Mathematical studies give the mind something certain to rest upon, while other studies lead it into a labyrinth of perplexity, bewilderment and confusion. The mathematician deals in certainties. There is no doubt, no mystery, no ambiguity, no variation, in the great principles which govern him; he feels that they are under the immediate control of Him, with whom "there is no variableness, neither shadow of turning." Uncertainty belongs to everything else, history, philology, metaphysics, geology, chemistry, etc.; "the trail of the serpent is over them all," the father of lies has tarnished everything bright and beautiful about them. But the spirit of immutable truth presides over all the investigations of the devotee of science. He can arrive at the same great results, not only by a hundred independent routes, but by a thousand by-paths. Thus, there are more than a hundred direct demonstrations of the square upon the hypothenuse, and yet the truth of the theorem may be made to appear as a consequence of innumerable other propositions in geometry. It can be deduced, for instance, from the properties of similar triangles, and from the area of a triangle in terms of its sides. So, too, the truth that the two tangent lines, which can be drawn to a circle from a point without, are equal in length, may be shown also from the relation of secants to their external segments, or from the relation of a tangent to the whole secant and its external segment. Likewise, the measure of the surface of a sphere may be deduced from that of a zone, and the measure of a solid sphere from that of the spheric segment. Similar remarks may be made of every proposition in Euclid or Legendre. But not only do mathematical truths admit of innumerable direct and indirect demonstrations, in the particular branch to which they appropriately belong; they are moreover confirmed and verified by every other branch of the science. Thus, in arithmetic, the product of two numbers, whose sum is fixed, may be shown to be the greatest possible, when the numbers are equal. Suppose the sum of the numbers to be 10, the numbers themselves may be 1 and 9, 2 and 8, 3 and 7, 4 and 6, or 5 and 5; and it will be seen that the product of 5 by 5 is greater than the product of either of the other sets. The same truth is more rigorously demonstrated in Algebra; it has been elegantly proved by Hutton in his Isoperimetrical Geometry, and by Newton in his Maxima and Minima, and has been used by Vauban in determining the proper form for field and permanent fortifications. Some 250 years ago, |