citing astonishment by the novelty of the fact which it communicates, and pleasure at the extraordinary beauty of which its demonstration is capable. But when, on a further progress in science, and very little is requisite for the purpose, its signal utility in suggesting, demonstrating, and accumulating, new truths is perceived; when we find its applicability to the business of life, and its omnipresence as it were, either palpable or latent, in the Arts; we are inclined to regard its practical utility, as well as scientifical importance, in an exalted point of view. The 37th and 47th are deservedly considered as the brightest stars in the zone of Geometry.

It is to Pythagoras, the "Samian sage," that we owe the discovery of both these theorems; but whether as an inventor, or a transplanter from other climes, is not certain. We can hardly suppose the Egyptians to have been ignorant of such remarkable truths, or the Chaldeans, who made considerable advances in Astronomy. That the Chinese have been for many ages in possession of the 47th, and a mode of demonstration peculiar and national, is strongly attested. The well-known story of Pythagoras, that upon his discovery of the 47th he sacrificed a hecatomb to the Muses, would, if true, be decisive in his favour; but besides that the same tale is related of Thales with regard to another proposition, it is disproved by the fact of his poverty in the first placet, and his religious creed in the second, it being against the doctrine of transmigration to spill animal blood.

The practical uses and applications of this theorem will be seen chiefly in union with some elements delivered in PART III., for the COMPANION of which we shall reserve them, with a few exceptions, which may be enough at present.

A man has two squares of building ground, ABCD, defg,

The 32d of Euclid's Elements.

+ It should be considered, however, that this argument has little weight, if we suppose the sacrifice not to have been performed till the philosopher's removal to Italy, where he enjoyed great power, if not riches. But the other objection appears, itself, sufficient to discredit the story.

I

neither of which is large enough
for the house he wishes to erect.
His neighbour offers to give him
in exchange a square, AHIK, out of
his ground. How shall he know
if he gets as much ground as he gives?

K

Let him measure the distance AE, and if this be equal to AH, the side of the square offered, then the grounds exchanged are exactly equal in quantity. For, by ART. 33, the square described on AH is equal to that described on AE, as these lines are equal; and by ART. 47, the square described on AE is equal to the sum of the squares, ABCD, DEFG, as these are respectively the squares described on the three sides of the right-angled triangle ADE. Consequently the square AHIK is equal to the sum of the squares ABCD, DEFG.

This method may be adopted even when the three squares, ABCD, EFGH, and IKLм, are separate; by producing EH (the side of one of the lesser squares) until the produced part, EN, be equal to AD (the side of the other

lesser square). Then, if the

K

E

B

D

straight line NF be equal to AK (the side of the greater

square), the exchange of ground is equal.
described on EN is equal to ABCD.
Again: Let the breadths, AN, BM,
of two bales of cloth, be stretched
perpendicular to each other at their
corners, N and м. Let the breadth
of another bale stretch exactly from
A to B. Now, if pieces containing
the same number of breadths be
cut from all, one of these pieces
will be equal to the other two to-
gether.

H

For, the square

Λ

N

M

B

с

For: if BC, ND, AH, be respectively breadths of the three bales, then BCFM, NDGA, AHEF, are squares; and as ANB is a right angle, the square AHEB is equal to the sum of the squares BCFM, NDGA. Likewise,

if successive breadths be taken on the three bales, the second square on the broader bale will be equal to the sum of those on the narrower bales; and so on. Hence, if the same number of breadths be taken on all, the pieces cut off from the broader bale will be equal to the other pieces taken together, as each successive square of the former is equal to the sum of each successive pair of squares of the latter.

THE

GEOMETRICAL COMPANION.

PART II.

IN the COMPANION to PART I. the " uses and applicacations" of GEOMETRY were exemplified with respect to the elementary principles of Angles, Triangles, and Parallelograms. Limited as we are by the size and scope of our work to brief and simple illustrations, PART I., containing only the rudiments of this Science, was peculiarly suited to the ends we had in view. As we advance to higher scientifical principles, the number of familiar examples by which they may be practically set before the eyes of the reader must necessarily diminish. To understand their uses and applications, he must understand a great deal more of the Arts and Professions of life, than was required in PART I.; which is not to be expected from our readers. Or, in order that he should understand so much, we must enter into details of those Arts and Professions, which would be totally incompatible with the nature of our Treatise. Nevertheless, PART II., as it is but one step beyond the very foundation of GEOMETRY, will be found by no means deficient in principles of obvious utility, and of almost universal agency in the common affairs of industrious life. Indeed, if we reflect a moment upon the geometrical figures, whose properties are developed in this Part, we shall see that by their resemblance to those which continually meet our eyes in the works of Nature and of Man, there must be an intimate connexion between them; and that a scientifical investigation of the former will,

therefore, disclose many facts with regard to the latter, not only interesting and curious of themselves, but fruitful in advantage of a really solid kind when brought to bear on practice.

The Circle is a figure with which we become early familiar, and which innumerable objects daily present to our senses, of a more or less regular description. The face of the Sun (called the disc), is always a circle, and besides being so constantly before our eyes, is perhaps as close an approximation to a perfect, or mathematical, circle, as a material object in nature can furnish. The full Moon presents a disc little less accurately circular to the naked sight; though when viewed through a powerful glass, its boundary appears indented and uneven. The Planets exhibit discs, which are greater or lesser segments of circles, according to different circumstances to be explained in Astronomy. Fixed Stars, likewise, have their discs circular, but of so incalculably small a surface at their great distance from the Earth, that they afford rather an idea of a point than of a circle. Again: the celestial paths which these bodies tread, or appear to tread, are either circles, or curves nearly such. If the courses apparently trodden from east to west every twenty-four hours were traced on the heavens, they would be circumferences, or parts of circumferences, lying parallel to each other, and whose diameters gradually diminished as each lay nearer the pole-star. Those other

paths, which Astronomers have now discovered to be real, namely, the planetary Orbits, or the curves trodden by the planets round the sun, though not exact circles, approach that figure so nearly in most cases, as to be considered such, where great nicety is not required. Descending to Earth itself, we find this body of a globular form; so that if sliced through and through, the flat face, or section, of each part would be circular; at least nearly so. Rainbows are splendid illustrations of our position, being generally arches of a circle resting upon the ground. Many natural forms and productions of Earth exhibit circular appearances, either in their outlines or sections. Hence we may perceive, that in giving a scientifical theory of the Circle, geometry is not to be considered as merely indulging a fanciful speculation upon what had no existence in Nature; as