gaged the attention of the Pythagorean school, and they also determine the utmost limits to which the science had been advanced by the united labours of geometers during 200 years after its introduction into Greece.

To the Pythagorean succeeded the Platonic school, the institution of which forms an epoch of great importance in the history of geometry. By the successful efforts and ingenuity of this school, its boundaries were greatly extended and new and attractive subjects of discussion introduced. In it originated the conic sections, the geometrical analysis, the geometrical loci, and the discussion of the celebrated problems of the duplication of the cube, and the trisection of an angle, the solution of which long exercised the patience and the mental energies of the ancient geometers. Plato himself was both a distinguished philosopher and an ingenious geometrician, as his invention of the geometrical analysis fully proves, who by his instructions and example created and cherished in the minds of his followers a fondness for the study. Finding that their predecessors had nearly exhausted the departments of plane and solid geometry as then known, they looked around. for new subjects for the exercise of their talents, and were at last successful in attaining the object of their search, by supposing a cone to be intersected in different directions by a plane. If a cone be intersected from the vertex by a plane passing along the axis, the section will be a triangle; if cut by a plane parallel to the base, the section will be a circle; and these two figures belong to plane geometry. But if a cone be cut by a plane parallel to another plane touching the conic surface, the section will be a parabola; if cut by a plane inclined to the base, the section will be an ellipse; but if cut

in any other manner, the section will be an hyperbola;-the three curves which are usually termed conic sections.

The ancients regarded the properties of these beautiful curves as objects merely of interesting speculation, in the detection of which they exercised their mental powers, but they never applied them to any practical purpose. In modern times, however, these curves have been found of great practical value, as they have been discovered to represent the paths or orbits which the planets and comets describe in their revolutions round the sun as their common centre of attraction.

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The geometrical analysis is a mode of demonstration the very reverse of the synthetic, which is commonly employed in teaching the elements of the science. The former is the method in which researches for the discovery of geometrical truths are conducted; the latter, that in which these truths are communicated. The geometrical loci take their rise from indeterminate problems. If every point in a straight or curved line, and no other point whatever, is always found to fulfil certain conditions, that line is called the locus of the point. These loci were principally applied by the ancients to the solution of determinate problems, which they were frequently enabled to effect by the intersection of two loci which had formerly been determined. Such were the ingenious and abstruse researches which exercised the dexterity and talents of those eminent men whom the Platonic school produced.

Geometers next turned their attention to the determination of the lengths and areas of curves. The methods of demonstration which they had hitherto employed were amply sufficient for the comparison of triangles or circles with each

other, but could not be extended to the comparison of rectilineal with curvilineal figures. In treating of these the ancients found it necessary to introduce a new method of demonstration, that of exhaustions, the most subtile and ingenious of all their inventions.

To effect their purpose, they inscribed one rectilineal figure within the curve, and also described a similar rectilineal figure about it; and varying the form of these by continually increasing the number of their sides, these figures at last approached so near the curve that the difference between them was unassignable, and in such circumstances any property of the rectilineal figure which is independent of the number of the sides becomes also a property of the curve, and thus its length and area are determined. Archimedes, the greatest mathematician of ancient times, employed the method of exhaustions with admirable address in establishing most of his discoveries. This method, indeed, involves the metaphysical principle of the modern calculus, that of variation, on which the latter wholly depends. This is the utmost limit to which the geometrical researches of the ancients extended, and it is nearly as far as the science can be advantageously prosecuted by the methods of demonstration which they employed. In this state geometry continued for nearly 2000 years until the revival of learning in modern times.

In demonstrating the truths of plane geometry, the ancients permitted nothing to be done before the manner of doing it had been explained, except the simple operations required in the postulates, or what can be effected by the aid of the rule and compasses alone, and nothing to be assumed as true without proof, except the few general and incontrovertible

propositions called axioms, and prefixed by Euclid to his elements. On the definitions and axioms the whole structure of plane geometry rests. In comparing magnitudes one with another the ancients employed the principle of supraposition, or mentally applying one of the magnitudes to the other, and by a process of reasoning evincing their equality or inequality. To this was added the method of proportion, which may in truth be regarded as an extended application of the same principle. By the method of proportion, magnitudes are supposed to be divided into parts each equal to some unit of measure of the same kind, and by a comparison of the number of parts contained in each the proportional relation of the magnitudes is easily ascertained. One magnitude may be contained in another a certain number of times exactly, or if the one is not an exact measure of the other, a third magnitude of the same kind may be found which shall measure both without a remainder. Such magnitudes are called commensurable, and may be expressed in numbers with perfect accuracy. There are many magnitudes, however, which cannot be measured exactly by any third magnitude, however small; such as, the diagonal and side of a square, which are called incommensurable, and which cannot be expressed in numbers with perfect accuracy, although they may be so to any degree of accuracy less than perfect. Euclid, by including both these kinds of magnitudes in the same definition, has rendered his reasonings on the subject of proportion so abstruse and subtile, that few students can be prevailed on to bestow that close attention and persevering industry which are necessary to acquire a thorough comprehension of his demonstrations, and consequently derive small advantage from the imperfect knowledge of them which

they usually obtain. Euclid's method of treating of proportion is therefore not well adapted to the purposes of education, and the greater number of students find it necessary to draw their notions of proportion from other sources which, although they may be less accurate, are more easily comprehended.

The ancient geometers, as already mentioned, made the principle of supraposition the foundation of all their geometrical demonstrations, and were thus able to rear the structure of the science on a secure and solid basis. They were careful to exclude the idea of motion from their reasonings, because they imagined that its introduction would impair the accuracy of their conclusions. Prosecuting the study of the science. on this principle alone, the ancient geometers by their successful efforts widely extended its domain, and brought it to such a state of perfection as to prepare it for the reception of those vast improvements and that extended range of application which it has received in modern times by another method of demonstration, viz., that of variation, on which the modern calculus entirely depends. Fluxions may be considered as the geometrical and the differential calculus, as the algebraical development of the same principle, and the results obtained by both methods are the same. Of the method of variation the idea of motion forms an essential element, for, if magnitudes alter their value, they must become either greater or less, both of which changes involve the idea of motion. All the demonstrations of modern geometry are founded on this principle, which is far more flexible and admits of a more extended range of application than supraposition. By the powers of the calculus the province of geometry has been extended to regions of inquiry of which the ancient geometers had not