of its principles; yet can there be a doubt that they would both have worked with more economy and accuracy, if they had understood the common properties of a right-angled triangle? And when they advance to the more difficult cases of spiral stairways and vaulted roofs, it is easy to see that an ignorance of principle may lead to both error and waste.

The difficulty in positive rules prepared for uneducated men is, that they can never bend to circumstances; and the workmen go on in a fixed track, in cases where they might have changed it without a variation of principle, but with the greatest economy of time and money. The calculation of the strength and stress of timber, though very simple in itself, is, notwithstanding, an analytical problem, which one unacquainted with the principles of algebra could not solve; yet is it everywhere important that it should be properly determined. Very recently the roof of a large cathedral in England, which was supposed to be a model of architecture, fell by its own weight, destroying in a moment the result of a great expenditure of time and money; a fact which could never have occurred had the architect resolved a practical problem in the strength of materials. In the construction of groined arches, whether for roofs, door-ways, vaults, or bridges, the principles of descriptive geometry are equally applicable and necessary; the catenary and elliptical curves, which are their best form, cannot be understood without the higher geometry; the arch cannot be built, without the greatest ex

travagance in the use of materials, unless the precise form of every stone is known before it is cut from the rock. Such was the fact in some of the finest specimens of modern architecture; and such also was the case in the building of Solomon's temple; for it is recorded in the book of Kings* that "the house, when it was building, was built of stone made ready before it was brought thither, so that there was neither hammer nor axe, nor any tool of iron, heard in the house while it was building." And this fact also corroborates a former position, that geometry long preceded analysis.

If the quantity of timber, stone, and other material wasted in building, from the want of a very little knowledge of mathematics, could be calculated, I have little doubt its price would educate all the young mechanics of the land. Science is economical; it repays the people a hundred-fold for what is expended in its cultivation.

Let us take another example in the case of surveying. Everybody knows that carrying the chain and compass, and blazing trees, is no very difficult operation; yet of what USE would it be, if there was not mathematical knowledge to calculate the results? The surveyor himself must at least have some knowledge of trigonometry; and is it not obvious that every chainman in the forest would perform his duty better, if he were acquainted with the objects and

* 1 Kings, vi. 7.

principles of the business in which he is engaged? He would then know where and how to apply his labor to the best advantage. But in the beautiful survey of the Northwestern Territory, mathematics has exercised a still higher faculty; all the sectionlines are based upon the meridian-lines, and these meridian-lines were fixed by the nicest astronomical calculations, while yet the Indian had not learned the mastery of the pale-face, and civilization announced itself only in the triumph of its proudest sciences!

In hydraulics we find the principles of mathematics equally necessary: here, all the calculations of the velocity, power, and quantity of moving fluids depend upon these principles. How can a millwright be master of his business without understanding them? The very shape of the cogs in his wheels are determined by them; their form is that of the cycloid -a curve generated by a fixed point in the circumference of a circle revolving in a right line; and he must understand that curve, or he can never judge whether his wheels are fit for use. And how is he to ascertain the quantity of water necessary to move them? and how is he to ascertain the quantity discharged? If he will turn to a practical treatise on mills he will readily find a rule for it, but one which neither improves his understanding nor his pocket. If, however, he would study a few of the laws of forces, of descending bodies, and moving fluids, he could then make a rule for himself, and could adapt

it to all the changing circumstances of locality and

power.

In the construction of canals, railroads, bridges, and in all the operations of civil engineering, mathematics are the essential element. In addition to algebra and geometry, trigonometry and the conic sections find him full employment. Mountains and valleys are to be reduced to a level, rivers turned. from their channels, and all to be done with a certainty and economy which nothing but the calculation and reasoning of mathematics can effect. And when the beautiful and grand result is obtained; when the high hills are brought down and space traversed with the speed of the winds; when the people and products of the most distant nations meet together with the ease and safety of near neighbors ; when knowledge is borne over the earth by the chariot-wheels of all-conquering science; when civilization and Christianity herself look to these results as their kind and beneficent aids; shall we not inquire by what means they were accomplished? Shall we learn nothing from the principles by which this vast machinery is moved? Or shall education neglect them, when she is gathering the elements of a great and useful mind? Of the millions who rejoice and wonder and admire over these achievements, few are either taught or seek to know the means by which they are produced. Genius, cries the assembled multitude, genius is great and glorious! Yes; genius is indeed great—the admirable work of a perfect being :

yet genius unaided has done none of these things; but with industry and vigilance she has gathered the aggregate wisdom of uncounted ages; she has called arithmetic from the land of Chaldea; geometry from the plains of ancient Greece; logarithms from the hills of modern Scotland; and from the darkness of deep antiquity, as well as from the brightness of the fresh and living present, she brings the treasures of science to aid her in blessing mankind.

The connection of mathematics with the arts, sciences, and employments of civilized society are far too numerous for reference here. Those I have selected are cases of ready and familiar observation : and if they enter thus into the accustomed walks of life, still more do they into those higher and nobler studies, whose object is to develop the laws and structure of the universe. Man may construct his works by irregular and uncertain rules; but God has made an unerring law for his whole creation, and made it, too, in respect to the physical system, upon principles which, so far as we now know, can never be understood without the aid of mathematics.

THE STUDENT WITHOUT MATHEMATICS.

Let us suppose a youth who despises, as many do, these cold and passionless abstractions; yet he is intellectual-he loves knowledge; he would explore nature and know the reason of things; but he would do it without aid from this rigid, syllogistic, measur