having arbitrarily created a figure for its own abstract purposes, without taking the least hint from its resemblance to real objects, or having any views of practical utility in developing the properties of such a figure. On the contrary, it was frequent experience and observation of a certain form, which we now call circular, in visible and tangible objects, actually existing, that first suggested a scientifical theory of the Circle. But even though we granted that this theory had its origin in the whims of genius, or the visionary excursions of imagination, so numberless are the natural objects and things to which it applies, that it could not have been better adapted to explain their properties, if it had been invented expressly for that purpose. The student's own memory will supply him with many more instances of natural circles than we have above enumerated; and consequently, if he wishes to understand Nature throughout these her various details, he must previously understand the scientifical theory of the Circle.

But it is in the works of Art that this figure is most frequently seen, under all its modifications and dimensions. Churches and Public Edifices, of almost every kind, present it continually to our notice, in one part or other of their structure. Theatres and Amphitheatres are for the most part only circular layers of brick or stone, rising gradually from the foundation to the cope. Towers, Spires, Columns, &c. are frequently little more; but that in some of these objects the circles diminish perpetually from the base to the summit. Arches, Vaults, Bridges, Aqueducts, furnish obvious instances: Palaces, nay, even ordinary Houses, from the mansion of the peer to the hut of the peasant, bear ample evidence of the ubiquity, as it were, which we have attributed to this figure. It offers itself to our attention in domes, windows, ceilings, doors; in decorations, ornaments, furniture, utensils; which are, with few exceptions, either circular or parallelogrammatic in their contour, their sections, or their subordinate features. In the Fine Arts, Perspective, Plan-drawing, Modelling, &c., but especially in Architecture, the circle is constantly employed, either directly or indirectly. With respect to the works of Art, may be said of Machinery as it is of Astronomy (which is indeed otherwise designated Celestial Mechanism), with

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respect to the works of Nature, that the circle intermingles its character with the whole systems, giving to each its regularity and beauty, as far as the eye is a judge of them. Many of the curves in both are, indeed, not accurately circular, but they are, for the most part, such near approximations to that form as to be scarcely distinguishable from it. What are Manufactories but combinations of wheels and axles, pulleys and tangent strings, circular rollers, and arms with circular motions ?-Little else. What is it first seizes on our attention in a Mill or Brewhouse? Water-wheels with boundaries of buckets or dashers; toothed-circles playing into one another, and communicating motion; grinders, in the shape of small rolling-stones, or flat cylinders. Nay, let us descend to ruder mechanism, to domestic mills and manufactories,-the common knifegrinding and the spinning-wheel. Here the circular form still offers itself to our contemplation; and we must acknowledge the necessity of thoroughly understanding the nature and properties of that figure which enters so largely into all human occupations. The Turner, Clock-maker, Watchmaker, might be almost all denominated circle-makers; for almost all their operations might be reduced to this onegiving a circular form to wood or metal. Wheelwrights, Carpenters, Joiners, Carriage-makers, Bricklayers, Masons, in order to deserve any of these names, must be conversant, practically at least, with the doctrine of the Circle. Artitisans of all descriptions, from him who constructs a moving tenement for a thousand men, to him who fabricates an engine for viewing mites or atoms,-from the Shipwright to the Optical Instrument-maker,-bear ample testimony by the works of their hands to the universal predominance of this figure amongst the productions of human ingenuity and labour. We cannot look at a Mariner's Compass, through a glass Lens, nor into a Spherical Mirror, without recognising in its form one of the most general species of circumscribed space which the long details of Art

exhibit.

From the preceding examples of what we have called the "ubiquity" of the Circle in all visible works, Divine and Human, it is almost needless to infer, not only the propriety, but the necessity, of investigating the nature of

that figure scientifically. Nor will any one be disposed to question the relevancy of such an investigation to affairs of real life, exclaim against it as a mere waste of time, which should be employed on something more practical than curious, nor sceptically inquire-Of what use can the study of geometrical lines and figures possibly be? Every candid person who has read the foregoing PART will allow the utility of this to be, at least, probable; and what we have just said in our prefatory observations will, perhaps, induce the belief that it is certain. To be convinced of this, it is only necessary for him to proceed through our first LESSON.

One additional circumstance we beg leave to impress on the student's recollection: Geometry has uses, practical uses; Geometry has applications, practical applications, which it is impossible in such a Treatise as this to explain, or even to enumerate. Moreover: Geometry is a handmaid science; by its principles are established, wholly or in part, those of other sciences, such as Optics, Mechanics, Astronomy, Navigation, Pneumatics, Hydrostatics, &c. &c. Now, it is indisputable that these sciences are of a practical nature, are of real utility in human life, and are of vital importance in worldly matters. Hence, it is demonstrative that Geometry, even though we could not exemplify its immediate bearing upon the useful Arts and occupations, must have indirectly, if not directly, a reference to practice; and that the benefits derivable to man from these secondary sciences, are in a great measure due to it, primarily. That a knowledge of Geometry is indispensable towards the attainment of the other Sciences the reader must believe on our assertion; or, if he doubts it, let him open almost any scientifical work, and observe the frequent reference to geometrical figures and calculations. With this circumstance in mind, even though he may look upon the simple and familiar examples, to which the elementary nature of our Treatise confines us, as far from sufficiently warranting the high eulogiums we have passed on this Science, he will have little remaining doubt of its merits and extensive utility.

DEF. XIV. " A Circle is a plane figure bounded by one line such, that all right lines drawn from it to one and the same point are equal to each other."

By ART. 48, it is proved that this point must fall within the bounding line.

DEF. XV. "In a circle the bounding line is called the Circumference, and the point to which the equal lines are drawn from the circumference is called the Centre."

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It is by the principle of these definitions that a Wheelwright, a Millwright, a Clock or Watchmaker, in short, every artist employed in the construction of wheels, regulates his operations. He makes a set of spokes, CA, CB, CD, CE, CF, CG, CH, CI, all equal to each other, and of the proper length. He attaches these spokes to the nave, or circular block, c, in such a manner that their outer extremities, A, B, D, E, F, G, H, I, may still be equally distant from the central point of the nave. He constructs the several pieces of the circumference, or felloe, all equally deep or thick, so that, when attached to the spokes, the whole series will form an exact circle, ABDEFGHI.

LEARNER. This is plain enough; but surely it does not require the aid of Geometry to know how we should construct a wheel, however delicate? Had the science never existed, carriages and clocks would have gone just as well, -at least, so far as their wheels are concerned. A boy of five years old will bend his hoop into a circle, that it may roll evenly.

TEACHER. You are still, I see, brimful of one of the most destructive prejudices with which a false system of philosophy has succeeded in confounding the intellect of man: a prejudice from which I vainly hoped what was said in PART I. had completely emancipated you. Geometry is not a sublime fiction of the brain; it is not an Utopian theory, built on speculative opinions and abstract contemplations; it is not the creature of arbitrary thought, -but of worldly experience. It is the child of Common Sense by nature, of Philosophy only by adoption. I repeat once more-there is a natural Geometry in the human mind, of which the scientific is only a clearer, a fuller, a more regular and comprehensive system. Pray recollect this: and do not hereafter tell me, when you find an exact agreement between natural and scientific Geometry, as in

the above example of your ignorant and my informed wheelwright, that the latter is superfluous.

LEARNER. I see that the latter is, in truth, the former, -only systematized and pushed to a far greater extent.

TEACHER. It is, I grant, in some cases purely abstract, and of no use whatsoever, except as an exercise of the understanding, or a luxury to curious imaginations. But with such as this I shall rarely trouble you. The following practical example may, perhaps, deserve your notice, if you are not still resolved that an Art, in order to be founded on Geometrical principles, should be totally distinct from those by which common sense would teach us to direct our operations.

LEARNER. I will endeavour to forget so ridiculous a prejudice.

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TEACHER. Suppose that a Turner wishes to make the flat surface abcdefg round. He fixes its centre c on the axle of an instrument called a luthe; which axle, by pressing a footboard, he causes to revolve with great rapidity. He then brings the edge of a straight blade, or chisel, ch, to touch the revolving surface; and holding it steadily in a horizontal position, first the corners, a, g, f, and then all the edge of the surface farther from c than c will be grazed off. By this means the figure is soon reduced into a circular form, of any required dimension, yaz. For: the cutting instrument being held at the same distance cx from the point c, while the axle performs one round, the bounding line yaz must at every point have this distance from the point c; and is therefore a circle, by DEF. XIV.

ART. 50. "Aright line perpendicular to a chord through its middle point will pass, if produced, through the centre of

the circle."

Windows and doorways are often constructed with a circular-arched top. If, for example, ad, bc, represent the jambs of a doorway, and that it be desirable to have the top a circular arch, with its two extremities resting on the jambs respectively, we may find the centre from which that arch is to be described thus: Divide

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