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ingly the author has carefully discarded from Geometry all that which properly belongs to Arithmetic; while those parts which relate to measurement, and thus really pertain to both, he has treated separately, delineating as precisely as

he could the demarcation between the two sciences. He has only admitted a part of the Theory of Proportions, and has added it as a purely arithmetical digression to the Introduction, for the sole purpose of uniformity. He has also scrupulously refrained from adverting to the applications of Geometry to Sciences, Industries, or Arts, as the mere mention of these numberless applications would have irretrievably obstructed the path of pure Logic, and concealed the true track from the dazzled eye of the student.

of the student. How can the applications of Geometry to Mechanics, for instance, be shown otherwise than by means of a complete treatise on that science ? The same may be said of those two important auxiliaries to Civil Engineering, the arts of Surveying and

· The question of incommensurable quantities, which has been the subject of such endless discussions, is a purely arithmetical

The ratio of two geometrical quantities may always be replaced by that of two straight lines, which have at least, according to the author's theory, their infinitely small elements as



common measure.

Levelling. Both of them are, so to speak, the most palpable forms in which Geometry is applied in practical life: still, both must needs be treated on their own merits, and quite independently of their theoretical substratum. The subjects of study in Geometry have a characteristic feature, owing to which the boundaries of that science may be defined more easily than those of any other; namely this that, being confined, as it were, to the human mind alone, their existence can only be foreshadowed to the organ of vision by means of graphic delineation, without being actually realized by the senses.

The first task devolving on the author was to determine upon the method best suited to his purpose. In Euclid's Elements, which seem to be a combination of the works of several authors, and a compromise between various diverging theories, the method is multiform, and often untraceable, owing to the very complicated nature of the system. But, in the treatment of any subject, it is essential that the method should ever be visible so as, like Ariadne's thread, to guide the inexperienced student throughout. Thus, it is most important to adopt a method which explains every part, and connects the whole naturally together in one body; and the best method will be that which does so in the most simple and lucid manner, no gap, no jerk, being allowed. It will be said that, for each subject, there is but one real method which can produce a perfect system, and unless that one true method be discovered, all efforts must prove nugatory. Now, perfection is not absolute, but relative; what is perfection to-day, will no longer be so to-morrow: it is a growing thing. The number of geometrical truths increases constantly, and a considerable addition to those known at present may be expected in the future, when the icy spell, under which Geometry is now lying dormant, shall have been broken. Every new discovery tends to exhibit more clearly the real connection, the Causulnexus, as the Germans term it, existing throughout the whole array of geometrical truths. Their ensemble has been compared to a chain ; a tree would be a more suit

l able simile. Geometers discuss the connection of the boughs with the branches, and of the branches with the trunk; and what is looked upon as an integral part of the trunk is often proved, by a more efficient method, to be but a secondary bough; but, let the method be ever so skilful, it can never reach the root or principle, except through the trunk which must consequently be clearly marked at the outset; and the more clearly it exhibits the whole system,-root, trunk, branches, and boughs,-their connection and their dependence on one another,—the nearer it comes to Nature itself.

A strictly logical system of Geometry has for - each proposition a fixed place, which it is the task of the geometer to discover, and which cannot be altered either to facilitate a demonstration, or for any other purpose. The frequent deviation from this rule seems to be the chief defect of Euclid's Elements; and this drawback must

} exist in all systems which are the result of a compromise. Upon the logical plan, the propositions must be arranged, according to the laws of the human mind, synthetically, from the simplest to the most complex, or analytically, from the complex to the simple, but always naturally, without disproportion or distortion of any kind.

The author refrains from going at full length into the principles on which his system is founded, because, on the one hand, these will be readily understood by a perusal of his work; and on the other, because the present treatise is designed but as a transition stage between the Euclidean plan and the author's ideal of a system of Geometry, which, he fears, would have been thought prematurely novel to-day, but which he still hopes to publish at some future time; and again, because he feels that his labours, if successful, will be


only preparatory to the great crisis which must soon settle Geometry as a science, when the time and the man are come. It is enough that he should now state that, however much he has departed from Euclid in the general arrangement of The Elements, and in many points of detail, he has still endeavoured to preserve throughout the main characteristics of Euclid's method. He has striven to attain that unity and plasticity that are wanting in Euclid's system, and which are indispensable to its further development. Thus, although he has altered almost every one of Euclid's definitions, he has but slightly modified his definition of the straight line; and this very definition thus modified, he has made the cornerstone of his whole system. Above all, he has tried to preserve the extreme rigour of reasoning, the admirable accuracy of thought and expression, which are the immortal glory of the great Geometer; he has adhered to the spirit of Euclid's fabric, if not to its letter.

The often repeated praise given to Euclid, that his book is, after two thousand years, still used by teachers in the exact form he has written it, must, the author thinks, hardly please his shade, as such praise is almost equivalent to the reproach of having stopped short the development of Elementary Geometry: the proof of life is progress.

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