deniable, as we experience in the commonest transactions of life, yet the utility of the doctrine on which they are founded is doubtful. This is as much as to say, that although the utility of a house to cover our heads is certain, yet the utility of the ground it stands on is disputable. We do not contend that it is as necessary to be familiar with the Doctrine of Proportion as the Rule of Three; no more than we contend for the necessity of a man's analyzing the nature of the ground on which his house stands, before he can live comfortably in it. But this we assert that it is impossible to deny the utility of a doctrine on which a Rule so confessedly useful is founded; just as it is impossible to deny the utility of a plot of ground, on which the house a man lives comfortably in is constructed. To be sure, the inhabitant knows that his house is built on the ground beneath it, and accordingly he respects the said ground for its manifest utility and convenience. On the other hand, a reader may not know that the Rule of Three is built on the doctrine of proportion: but is the said doctrine to be despised because the said reader is ignorant? Or are the merits of this PART of GEOMETRY to be considered problematical, because the qualifications of that reader to appreciate them are so? What can be the possible use of such things as "multiples," "submultiples," equi-submultiples," " ratios," and "proportions?" will a knowledge of them make us either wiser, or better, or happier? in short, is there any thing but abstract and visionary enjoyment to be derived from such high-flown speculations, in which this doctrine of proportion threatens to involve us? We have already answered these, and all such half-contemptuous, sceptical, superficial inquiries. Does the interrogator admit that there is any possible use in The Golden Rule? Does he admit that a knowledge of it is calculated to make us either wiser, better, or happier? In short, does he admit that there is any thing but abstract and visionary enjoyment to be derived from the speculation in which it involves us ?—If he does, then he likewise admits the same of the Doctrine of Proportion, on which that Rule is founded; if he does not, we would in our turn interrogate him, Does he admit any thing-his own existence, for example?

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But it is not only as a step to the knowledge of other sciences, however practical, that the doctrine of proportion should be regarded as capable of advancing the great end of life in a worldly sense-utility. In the various Arts and Manufactures, in Mechanics and Machinery, we shall find the advantage, if not the positive necessity, of this doctrine being clearly understood by those who would polish the rudeness of first inventions, and adapt them to produce, with the least expense of time and labour, what is requisite for the convenience and enjoyment of a highlycivilized state of society, to which every kingdom naturally verges. While nations remain in the thraldom of ignorance and barbarity, their arts and manufactures are few and inelegant, their mechanics and machinery are clumsy and comparatively impotent. These defects are almost solely attributable to the unacquaintance with GEOMETRY. When they begin to emerge from a condition so lamentable, it is demonstrated by an improvement in these four ticulars: their arts and manufactures become more numerous and refined, their mechanics and machinery become neater in their several details, and more energetic in producing their effects. In fact, the arts, manufactures, mechanics, and machinery of a nation may be looked upon as a kind of thermometer by which its rank in the scale of civilization may be estimated; or at least, of what is much more important,-its prosperity, so far as that is dependent on wealth and the presence of all the comforts, conveniences, and luxuries that wealth can purchase. But this improvement in the condition of society can be obtained in no other way than by a preceding improvement in the state of geometrical knowledge. It will be found that there is an intimate and indissoluble connexion between art and science in all these particulars. If we examine the enginery of a nation in this advanced state, we shall find that its superiority to that which was in use among the primitive inhabitants arises from the adoption of certain principles of geometry which were unknown before, by which its construction is regulated; so that defects are avoided, and new powers bestowed, according to the degree of geometrical knowledge embarked. But, perhaps, in this state it will be found that the geometrical knowledge embarked in the

construction of the national enginery rarely goes beyond the doctrine of Equality, and that the principle of equal lengths, areas, &c. is that which regulates the form of all implements familiar to that society. We may consider the introduction and application of the doctrine of Proportion, for the purpose of improving enginery, as the test of a nation having passed the boundary of daylight and twilight which separates the second from the third and highest known state of civilization. When this doctrine pervades the arts, manufactures, mechanics, and machinery of a nation, we may safely pronounce that nation to be in the enjoyment of those things which are held to constitute happiness in its worldly acceptation, and which it is no irreverence to say will justly be thought to form no inconsiderable portion of human felicity, while men continue to be mortals.

In the examples which we have collected to illustrate the method by which the doctrine of proportion is applied to practice, and all of which are selected of as familiar a kind as possible, there will, we would hope, be found proof sufficient of the assertions made in this preliminary section. But we are happy in being able to adduce a much higher opinion than any we can pretend to give on the merits of this branch of geometry, in at least one profession of an entirely practical nature,-Architecture. Speaking of architectural beauty, Sir Christopher Wren says:-" There are natural causes of beauty. Beauty is a harmony of objects, begetting pleasure by the eye. There are two causes of beauty, natural and customary. Natural is from GEOMETRY, consisting in uniformity (that is, equality) and proportion. Customary beauty is begotten by the use of our senses to those objects which are usually pleasing to us for other causes, as familiarity or particular inclination breeds a love to things not in themselves lovely. Here lies the great occasion of error; here is tried the architect's judgment: but always the true test is natural or geometrical beauty." It was by keeping perpetually in view this test, this geometrical beauty, which consists in uni

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formity and proportion, that the above illustrious Architect, who was also a profound mathematician, raised such a noble monument to his own glory and that of England, as Saint Paul's Cathedral. By the same means was its magnificent prototype, Saint Peter's at Rome, rendered one of the wonders of the modern world, under the superintendence of Michael Angelo, at once, perhaps, the greatest Painter, Sculptor, and Architect, of the Christian ages, and whose threefold celebrity is in a great measure attributable to his scientifical genius and learning. These sublime and interesting architectural remains of antiquity which still consecrate the classic ground of Italy and Greece, as well as the beautiful specimens of pagan art which are to be collected from descriptions, drawings, and medals, bear witness that no other test was observed by the Ancients in the construction of their public edifices than that by which Wren and Michael Angelo so happily profited. Indeed, as the Ancients surpassed the Moderns in a love for Geometric Science, as they cultivated it with more exclusive attention, and, as a consequence, possessed more dexterity in its application,-we accordingly find that their Architecture still transcends by a great many degrees that of their successors, although formed on the same model and on the same principles. In the heights, lengths, areas, &c. of their edifices, there is a harmony between all the parts, subordinate as well as principal, from the column to the moulding, from the pedestal to the pediment, which incontestably demonstrates how far the Doctrine of Proportions had inbued those who designed them, how religiously that doctrine had been observed in their construction, and how superior they must ever remain to all edifices that propose to rival them, until the same doctrine be exhibited, with the same judgment, in practice. Much is, perhaps, due to the exalted genius and taste of the Ancients in this department; but a great deal likewise to their sedulous cultivation of geometry, and in particular of that doctrine now under review.

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LESSON XII.

ARTICLE 94. "Parallelograms which have equal altitudes, have to each other the same ratio as their bases."

This is the foundation-theorem on which the whole doctrine of Proportion is built, as far as it regards rectilineal figures. There is a perpetual recurrence to it in geometrical investigation, and even in practice it is of extensive use and application. By its means we are able to compare surfaces, so as to determine that they are equal or unequal, which might be done by the principles of PART I.; but, in case of their inequality, to determine, also, how much one of them exceeds the other. For example, an oblong Cistern ABCDEF, ten feet and a half in height, CD is emptied from the brim to within a foot and three-quarters XD of the bottom. By Art. 24, we know that the parallelogram CIGX is not equal to the parallelogram XGED (for otherwise their

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PROB. XV." To divide a given finite right line into any number of equal parts.”

TEACHER. If it were only for the benefit derivable from this Problem we should not be disposed to contest the usefulness of geometry.

LEARNER. I see that it must be frequently of great advantage in practice; having myself very often occasion to divide a line into a certain number of equal parts. But I do not see why you should give such a flattering character of this problem?

TEACHER. It is the basis of all rectilineal Measurement by means of graduated instruments.

A Ruler is a plain straight bar of wood or other material, either round or flat-sided. A Scale is a straight flat-sided bar of wood or other material, whose whole length is divided into regular intervals by marks on its surface. Scales are often denominated Rules; but it would, perhaps, serve the great end of clearness and distinctness in our ideas, if the latter name were appropriated to such instruments as are used only for drawing