Deeply interested in these studies, and impressed with their great utility, we had, indeed, for a considerable time lost sight of the precise object to which we had limited ourselves in the above Prospectus. Instead of pursuing the line there marked out, we had deviated from it into numberless by-paths, all leading to something curious or useful, but nevertheless deviations. We had entered upon a critical examination of every proposition in our GEOMETRY; and investigated the peculiarities, beauties, difficulties, in a word, the most circumstantial details of each. We had interspersed a variety of COROLLARIES and Geometrical DEDUCTIONS, which might at once serve to exercise the reasoning powers of the student, and perfect him in the theory of the Science. We had, moreover, prepared a Series of Demonstrations, on an entirely different plan from those in common use, for the service of those whom the abstract nature of the latter might disincline to the study of Geometry. In addition to all this, we had to accomplish the professed object of our work. But after having expended an immense quantity of time and labour in the above pursuits, we found that our Volume, if continued on the same scale, would form one at least treble of the prescribed size and cost. Besides, we were not called upon to be so much better than our word; it was sufficient, if we were merely as good. With the benefit of this late reflection we suppressed all the gratuitous parts of our work, and confined ourself to that which would fulfil our promise, but no more.

In the present Volume will be found the completion of that part of our Prospectus which regards GEOMETRY. The abstract principles of that Science, as delivered in

Vol. I. of the SCIENTIFIC LIBRARY, are here applied (wherever conveniently so applicable) to practice; that is, in a great number and variety of examples, it is shown how these principles may be, and have been, brought to bear on the actual business of life, so as to evince their utility in the different Arts, Occupations, and Pursuits of Mankind. This being done, we apprehend few will hereafter be found to inquire-What is the use of abstract Geometry? Or if they do, we hope few will be found. without a sufficient answer, after having read our Volume.

A brief History of the Rise and Progress of this Science, so interesting to all acquainted with its objects and nature, forms a convenient introduction to the principal subjectmatter of our Treatise.

Addressing ourselves to the general, not the scholastic reader, nor him deeply instructed by experience in one or other branch of Art, we have constantly endeavoured to avoid all technical terms, all illustrations which imply knowledge beyond what common observation must have furnished, all details which might not be simply and shortly stated, easily and quickly comprehended. This, it is true, occasioned some difficulty in the assortment of proper materials, especially as we found but little assistance in that respect from works already published, even such as were professedly elementary and practical. The simplicity likewise of our plan, together with the conversational form, which we have adopted occasionally for the sake of giving a familiar interest to what is usually considered as repulsive, may be thought to throw an air somewhat too elementary over our treatise. But, as to this objection, which would be of weight if we wrote for the learned, it is of

none as we write for the unlearned reader. With respect to the trouble we may have had in assorting and arranging materials, it will be recompensed by a knowledge that it has not been undergone in vain; and this we hope, with a confidence derived from the approbation with which our other works have been received, to acquire from the liberality of an indulgent public.

THE

GEOMETRICAL COMPANION.

GEOMETRY, So far from being, as is generally imagined, a science which owes its birth to the abstract speculations of philosophers, indicates by its very name that it was essentially practical in its origin, was suggested by practical considerations, and cultivated at first for solely practical purposes. The word GEOMETRY, in its primitive acceptation, signified measurement of the earth; and is derived from two Greek words ge (earth), and metron (measurement). In fact this term was applied to an Art which more resembled that of Surveying than any other with which we are acquainted; being however much less artificial in its operations. From what circumstance it took its birth writers are not generally agreed. Herodotus, the celebrated Greek historian, attributes it to the intersection of Egypt by numberless canals in the reign of Sesostris, who flourished nearly fifteen hundred years before the Christian era, and who by this means is said to have partitioned the country under his rule among its inhabitants. The excavation of the lake Mæris in that kingdom, for the purpose of a reservoir, into which the overflowing Nile might discharge itself, is also affirmed to have given rise to this Art. But many writers upon the subject appear to consider the inundations of that river as affording the most probable cause for the invention and use of Geometry. The Nile annually overspreading the whole country, and thus destroying or effacing the landmarks which divided one man's agricultural possessions from those of another, rendered a subsequent partition necessary, by which the same quantity of ground as he had before the inundation

might be allotted to every person. Some principles of land-measurement were requisite to the accuracy and justness of this subdivision; and such principles are thought to have been the foundation, not only of the Art as it formerly was, but of the Science as it now is. It has been plausibly objected indeed to the account just given, that the supposition of the Nile effacing landmarks is gratuitous; for that the inundation was not in the manner of a torrent, but of a gradual swell, by which landmarks of any stability would have been left undisturbed. It is perhaps after all most reasonable to conjecture that this Art had its beginning in an age far more remote, and an origin far more simple, than any which has been assigned to it. The division of lands among children, or heirs of any kind, at the death of the proprietor, is sufficient to account for the introduction of some principles of land-measurement; which, few and rude at first, were gradually augmented and refined. As population spread, this division of landed property became more frequent, and therefore the principles by which it was regulated were more often consulted. They were at length embodied into a kind of Art, which was cultivated by regular professors, like our surveyors; and the first great work on record to which it was applied, such as the Canals of Sesostris, or the Lake of Mæris, gave it a remarkableness which has induced historians to take that as the era of its birth which was only the epoch of its maturity. It had existed long before, but some such work as the above-mentioned first brought it into public notice, and made it matter of history. Its true origin is probably to be referred so far back as to that age (the precise date of which is undiscoverable) when the human race had spread over too much earth for emigration, or the assumption of unoccupied land, to be an easy matter. Persons would then stay at home, and endeavour to sustain themselves on that portion of land which came to their share by inheritance. Hence land becoming thus valuable was necessarily divided amongst the co-heirs with the most jealous precision and accuracy; in which circumstance we find a natural source and origin of Geometry, as the art of landmeasurement, without recurring to the fables of antiquity for one that at the best is only possible.