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The necessity for a work, somewhat on the plan of the present, is the best excuse for an imperfect one, and the best recommendation of a good one. There is a Geometry (if it may be so called) for children, and a Geometry for professed mathematicians; but there is no Geometry for the public at large. We have nothing between Pinnock and Euclid. A Geometry as far beyond the one as behind the other, is the greatest desideratum in the world of literature.

Knowledge, as it exists now in the public mind, and as it tends to exist hereafter, requires such a work; in order to render that which is already gained more satisfactory, and that which is to be gained more accessible, than either can be without the aid of Geometry. This is the foundation of all scientific knowledge, as the study of it is the surest step to all Advancement in Learning of the philosophical kind. Without geometry may be obtained information of facts, or knowledge by rote, which is the sort now chiefly extant among us: But it is only by means of geometry that a scientifical knowledge of things can be acquired, in which we are now generally deficient.

A popular work upon almost every science has appeared: it is the design of the present to furnish a System of Geometry in like manner adapted for public use.

There are three classes of persons for whom it is especially if not solely intended : Youth at public and private Schools ; Persons whose Education has been neglected; Artists and Mechanics. To the first it will be an introduction to Euclid; to the second, a private and easy course of Elementary

Mathematics; to the third, a Manual containing all the geometrical principles on which the knowledge and practice of their several Arts are founded. We could point out a fourth class for whom our system of Geometry is peculiarly adapted: but custom, and illiberal prepossession against the ability of Ladies for studies of this nature, will probably ever prevent their deriving any advantage from a Science, whose very object is to strengthen the understanding.

As to the first class : It is well known by those who have the instruction of Youth, that the elements of Euclid is a work by much too large and difficult to constitute a proper school-book. Ere opened, its size begets fear, and when opened its abstruseness engenders disgust. Instead of rendering the principles of Geometry as familiar as possible, it renders them as abstract; not from an affectation of mystery, but from an anxiety for refinement. However desirable this may be in a philosophic point of view, it is highly inconvenient for the purposes of instruction. Beautiful as is the temple of Science, it lies so remote and inaccessible, that we must smooth and shorten the path to it, if we expect young and wavering dispositions to attempt the journey The first object, therefore, of our Treatise is, to supply an Introduction to Euclid, for the use of Schools, whereby Youth may be gradually familiarised with the notions of Geometry, and after learning half the Elements with ease in our volume, those who are destined to the University may learn the whole of them without difficulty in larger ones. The labours of acquirement, and instruction, will be thus respectively and equally diminished.

As to the second class: Many persons whose education has been, from want of time, inclination, or resources, neglected, in after years have the will and power to remove their deficiency. It is too late with such to begin the scholastic course of education, but the more limited and private one, furnished by scientific yet popular Compendiums, is within their ability. Likewise, there is a number of persons whose education, though far from having been neg. lected, was not of such a kind as the extraordinary advancement of knowledge, within the last few years, demands from those of their age or condition. If they have any acquaintance with the Sciences, it is superficial and altogether inconclusive. Many interesting truths which they meet with in their daily reading are unintelligible to them, many agreeable literary pursuits are shut against them. To this class of persons, also, a series of works like those described would be invaluable. But as Geometry is the key to all the Arts and Sciences, a popular treatise upon it is indispensable to those who would acquire the latter. That such a work should follow, instead of precede, the elementary treatises which have been published on Astronomy, &c. will not appear surprising when the abstract nature of the Science is considered. Our second object, therefore, is to furnish a Substitute for Euclid, from which persons whose education has been neglected, or restricted, may obtain as much geometrical knowledge as is sufficient, but not superfuous, in their circumstances.

As to the third class : Reasons similar to the above render Simson's Euclid (the only edition used in this country) totally unfit for the Artist or Mechanic, independent of its high price. Yet for no class of persons is the knowledge of Geometry more requisite. Every mechanical profession involves some principle of Geometry, and according as the Science is well understood so will be the profession, ceteris paribus. Every mechanical invention and improvement is · founded on a principle of Geometry, and according as the Science is ready to the mind, so will be the inventions. It is true that professions have succeeded and inventions originated with those who were ignorant of Geometry; but this happened not because of their ignorance, but in spite of it. Where theory can direct practice, and practice suggest

to theory, there is the whole man brought to bear; and what neither could do separately, both will do conjointly. Education has spread too far among this most useful and respectable class of people to leave our assertion either unintelligible or doubtful. It is therefore our final object to publish for their use a Compendium of Euclid, brief to suit their time, easy to suit their apprehensions, and cheap to suit their purses.

To accomplish this threefold object, the same course is sufficient. First, the principles of the Science must be rendered as familiar, and brought as near to our commonest ideas as possible. For this purpose we have omitted all technical terms which could be dispensed with; we have given popular illustrations of some things which were abstract; and we have simplified many doctrines which were perplexed.

The next step should be to render the demonstrations of propositions as plain for the mind, and as brief for the memory, as possible. For this purpose we have made several alterations : Instead of notes of reference, which distract the mind, we have introduced the Articles themselves referred to; instead of accumulating many theorems under one head, as is frequently done in Euclid, we have separated and proved them distinctly. Where the proofs in Euclid appeared difficult, unsatisfactory, or prolix*, we have substituted easier, clearer, and shorter ones ; and introduced other improvements of which the old methods were evidently susceptible.

Our last endeavour should be to reduce the Elements of Geometry, not only to their simplest, but to their shortest form. This was with us a principal object: conceiving it as unwise, as it is unphilosophical, to introduce as Elements of the Science propositions which are not such; thereby oppressing the reader's mind and memory with superfluous

* As in PROPOSITIONS 5, 29, 26, Book I., &c.

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