opposite) they attributed one of the same magnitude, except where the bust was a model taken from the life, as that of Cicero, in which the facial angle is only about 85°.

Here then we already see, in Mechanism, Craniology, and Sculpture, a few specimens of the intimate connexion between the abstract science of Geometry and the practical affairs of life. Did we proceed no farther than the mere definition of an angle,

we have in this ground enough whereon to establish the probability that a science which so early exhibits its close alliance with the various worldly occupations of men, cannot in its future details be very irrelevant to them. But proofs still more unequivocal than these will thicken upon us in the few next pages.

LESSON I.

ARTICLE 1. " If there be two triangles which have two sides of the one equal, respectively, to two sides of the other; and likewise the angles contained by those sides equal to one another: then the bases or third sides of these triangles are also equal.”

ART. 2. "In such triangles as above described those angles at the bases which are opposite equal sides, are respectively equal."

ART. 3. "Such triangles as above described are equal in every respect to each other."

LEARNER. I do not see what use or application can be made of the preceding Articles.

TEACHER. Four streets, AB, AC, AD, AE, equally wide from each other, meet at A; and another street, CB, crosses the two first, its ends, c and B, being equally distant from A. › It is designed to form a place, or square,

A

at A, all of whose sides shall be equal, and one of them the street CB: the positions of the other sides are determined by ART. 1.

For, taking AO, ap, as, at, ar, aq, all and each equal to am or an; then, the lines op, st, rq, will be the required positions for the other sides. Because, the four given streets being equally wide from each other, the angles man, oap, sat, raq, must be all equal; and as each pair of sides containing these several angles have been also taken equal, the bases, or third sides, mn, op, st, rq, must be equal, by ART. 1.

Again: ABCD represents the front of a line of houses seen in perspective, ADEF the end-face of the nearest. On the other side, GH, of the street (supposed to be viewed directly forward) the end-face, IKLG, of a house, exactly similar to ADEF, is represented. How is the draughtsman to draw

a line from L so that the range of houses on this side of the street shall appear under an equal degree of perspective with the former ?

Let him produce AB, DC, until they meet in м; let him also produce GH until GN equal AM; and then let him join LN. The front of the range GHOL will thus be seen under exactly the same degree of perspective as the front ABCD, by ART. 2.

For, since in the triangles GNL, AMD, the side GL was granted equal to AD, and GN is taken equal to AM; likewise, since the angle LGN is equal to the angle DAM (the cornerlines AD, GL, being situated similarly with respect to the street lines AB, GH); we have by that ART. the angle GNL opposite GL, equal to the angle AMD opposite the equal side AD of the other triangle. Hence, both ranges are seen under equal angles, that is, have the same degree of perspective.

Finally: Two sail-packets (rigged in the usual manner), if made exactly on the same model, should carry exactly

meets the mast in both vessels; and a rope stretched from the top of the mast to the point of the bowsprit. Then there will be two triangles ABC, abc, with the sides ab, ac, equal respectively to ab, ac, and the angle a equal to the angle a (the vessels being supposed exactly similar in construction). Consequently, by ART. 3, these triangles will be exactly equal; and hence will require canvas of exactly the same dimensions to suit them.

There are numberless other "uses and applications" of this tri-partite theorem, which the student will be able to recognise in the various works of art he surveys; but its chief use lies in this, that it is the principle on which still more practical elements are founded, as we shall immediately perceive.

LEARNER. I never considered the Articles in this light before. It is now plain that Geometry is by no means restricted to the development of the properties of figures drawn upon paper; but through their medium is extended to the materials we see or handle daily in our several occupations.

TEACHER. Yes; but its utility is not confined even to these particulars. You have mentioned the two first and most obvious uses of the science; it has two others almost as valuable.

LEARNER. What are they?

TEACHER. It teaches to reason systematically; that is, to build conclusion upon conclusion in a regular manner, so as that your last inference shall legitimately flow from

They seldom do so however: first, because they never can be made by human hands exactly on the same model; secondly, because the materials with which they are constructed and freighted are not in every respect precisely similar. But nevertheless a sailmaker should furnish the same canvas for both, and let it be afterwards adjusted to each by experiment.

your first position. The third, and perhaps the most essential use of Geometry is, that it teaches to reason strictly, and therefore soundly. Most errors in reasoning arise from negligent phraseology, and taking things for granted without sufficient examination. Now, if in Geometry we were to take for granted every proposition which seemed to be true on a mere superficial consideration of it, there is no doubt but that this science would be as infirm and unstable as any other system of reasoning.

LEARNER. I own that in some cases I should be as likely to think that the very contrary of the proposition, shown to be true, was true; for instance, that the three angles of every triangle were, taken together, neither greater nor less than one certain quantity. Had I not read the proof of this, I should be rather inclined to think that the three angles of a large triangle were greater, when taken together, than the three angles of a smaller one.

TEACHER. A celebrated writer observes, that the two best instruments for obtaining knowledge are-doubt and curiosity. By "doubt" he means a wholesome fear that what you take for knowledge may not be such. To sharpen this instrument is a signal use of Geometry. We are naturally, from the rash and presumptuous disposition of our minds, in want of some science which may teach us by habit to doubt our instantaneous judgments, to suspect our intuition, upon which, from idleness, we are apt to rely overmuch; and this science is Geometry. By accustoming us to expect the fullest and most exact proofs, even in the most obvious matters, it not only sharpens our faculties, corrects precipitancy, and prevents liability to imposition, either from the arguments of others or our own, but it gives us a satisfaction in our knowledge which we could not otherwise obtain, and thereby enables us to speak with a strength and perspicuity which a vague persuasion of the truth of what we utter could never inspire.

LEARNER. A knowledge of Geometry would be very useful to a public speaker.

TEACHER. There is no profession or pursuit to which it would not be so,-but that of a poet, whose imagination it might tend to restrict, and whose feeling it certainly could have nothing to do with. But wherever, not the reasoning

(which even a poet cannot do without), but the ratiocinative, or argumentative, faculties are employed, there must Geometry be of the first use. What I have said

above is the general answer to those who, being ignorant of the nature of Geometry, ask-Of what use is it? These persons seeing figures unlike any thing they commonly meet with, and hearing geometrical truths which do not seem to bear upon the ordinary business of life, but appear speculative and abstract, are ever ready to underrate what they do not understand. But they now understand this, that Geometry, if it does nought beyond, at least renders the mind more acute, watchful, and discriminative, in its search after truth; and this, we hope, is alone sufficient to recommend the study. But it is also the handmaid to the other Sciences,-Astronomy, Optics, Mechanics, &c., and must be learned before they can be acquired. It is the inventor, or improver, of almost all those Arts which provide the conveniencies and luxuries of life, and either directly, or indirectly, contributes to the happiness even of those who despise it. Perspective, Plan-drawing, Modelling, Carpentry, Masonry, Architecture, and Mechanical Arts without number, have their foundations in this Science. The invention of Geometry, as it is the greatest boast of the human race, so is it the greatest benefit to it. A bee, which is an instinctive geometer, finds the use of it in constructing its comb, so as to be the most roomy and durable at the least expense of time and labour. Will any one after this fact doubt the use of Geometry to an intelligent

creature?

LEARNER. I should like to know how the bee accomplishes what you have said; the fact is very curious.

TEACHER. Proceed then with diligence in your studies; you cannot expect to understand an operation which is highly geometrical in its principles without a knowledge of Geometry.

ART. 4. "A triangle which has two of its sides equal, has also its angles, opposite the equal sides, equal.”

LEARNER. From ART. 4, I see that both legs of a compass being equal, if both touch the paper, they must both form equal angles with the line joining their points.