us, does not excite any precise tendency, and which a mind capable of representing before it at once 36 objects or facts in a distinct state, would alone experience. By this artifice, we attain the same results as creatures endowed with imaginations and memories far more clear and vast than our own. Here, as before, all has been effected by substitution. After having enabled us to arrive at abstract qualities, she affords us the means of counting and measuring quantities. Thanks to substitutions, we were enabled to conceive the abstract qualities of individuals. Thanks to a series of repeated substitutions, we are able to name, and consequently to conceive, certain abstract properties peculiar to groups-properties which the natural limitation of our imagination and memory seemed to hinder us from ever conceiving, that is to say, from naming.

II. The efficiency of substitution extends far beyond this. As the reader knows, geometrical objects do not exist in nature. We do not meet, and probably can never meet, with perfect circles, cubes, and spheres. Those we see or construct are but approximately so.-Nevertheless, we conceive them as perfect; we reason about figures of absolute regularity. We know, with complete certainty, what is the obtuseness of each angle in a regular myriagon, and to how many right angles the whole of its angles taken together amount. Besides this, when for the better apprehension of a geometrical theorem we construct a diagram on paper, we trouble ourselves very little as to its perfect proportions; we admit without difficulty shaky lines in our polygon, and irregular curvature in our circle. In fact, we do not consider the circle traced on the paper; it is not the object, but the aid of our thought; by its means we conceive something differing from it, which is neither black nor traced on a white ground, nor of any particular radius, nor of unequal curvature.-What, then, is this object we conceive, and of which experience affords us no model? The definition tells us. A circle is a closed curve, all whose points are equally distant from an internal point called the centre. But what have we in this phrase? Nothing,

except a series of abstract words which denote the genus of the figure, and another series of abstract words which denote the species of the figure, the second being combined with the first, as a condition added to a condition. In other terms, an abstract character denoted by the first words has been joined to an abstract character denoted by the second words, and the total compound thus constructed denotes a new thing to which our senses cannot attain, which our experience cannot come in contact with, which our imagination cannot trace. There is no necessity for our attaining to, meeting with, or imagining this thing; we have its formula, and that is enough.

In fact, this formula would be rigorously the same if the object had fallen within our experience. We have constructed the formula before instead of after the experience, and it corresponds all the more closely to the thing, since the thing must bend to it—not it to the thing. The two then make up a couple whose second term, the definition, is equivalent to the first term, that is, to the object. -This object may remain ideal: it may itself be situated beyond our grasp; it matters little; we have its representative. Whatever properties and relations we find in the substitute we shall safely ascribe to the thing for which it is substituted. We arrive at this indirectly, as a surveyor, who, wishing to measure an inaccessible line, measures a base and two angles, and knows the first quantity by the three second.-In this way all mathematical conceptions are formed. We take very simple abstractions, the surface which is the limit of the solid, the line which is the limit of the surface, the point which is the limit of the line, the unit or quality of being one, that is to say, distinct existence among similar things. We combine these terms together and form, first, compounds of small complexity, those of two, three, four, and the earlier numbers, those of plus and minus, of greater and less, of longer and shorter lines; then those of straight line and curve, of triangle, of circle; then, those of sphere, cone, cylinder, and the rest. The complication of compounds goes on increasing; it is un

limited. Taken together they form a kingdom apart of objects which have no real existence, but which are capable, like real objects, of being classed in families, genera, species, and properties, of which we discover by considering in their place the properties of the formula which we substitute for them.

By a strange continuation the process which has formed these objects is also that which establishes their relations. Arithmetic, algebra, geometry, analytical geometry, mechanics, the higher calculus, all the propositions of mathematical science, are substitutions. Any number we take is a substitute for the preceding number added to unity. To calculate, is to replace several numbers by a single one at the end of several partial replacements. To solve an equation, is to substitute terms for other terms with the object of arriving at a final substitution. To measure, is to replace an undetermined magnitude by another magnitude defined in its relation to unity. To construct a diagram for the demonstration of a theorem is to substitute certain known lines and angles for other lines and angles which it is required to know. To find the algebraic formula of a curve, is to discover a mathematical relation between certain lines which are connected with the curve, and to translate quality into quantity.However we may reason about numbers and magnitudes, the process amounts to passing from one equivalent to another equivalent by the aid of a series of intermediate equivalents, to replacing magnitudes by numbers expressing them, a figure by a corresponding equation, a complete quantity by a quantity in process of completion, having the first as limit, movements and forces by lines representing them. We pass from each province to the other by substitutions, and, as a substitute may have substitutes, the operation has no limits.

III. Leaving for the moment this extension of our process, let us consider it once more at its outset. We have seen how, by combining abstracts, we construct the first terms of couples, the second terms of which are beyond our reach,

and how, by the study of the generating formula, we discover the properties of the object engendered by it. In certain cases, we discover in it wonderful properties, and the formula makes known to us facts situated not only beyond our experience, but beyond all experience.—If we divide 2 by 3, we find an infinite decimal fraction, 0·6666 &c., and we can prove that it is infinite. It is strictly so, and without possible break; however far we may prolong the operation, the remainder will always be 2 and the quotient always 6. After a million, after a thousand million, after a million million of such divisions, new terms will present themselves, with the same remainder and the same quotient, with a total quotient always too small, too small by a fraction with 2 for numerator, and for denominator unity followed by as many zeros as there are units in the number of divisions we have made. Here is something infinite; not vague, not indefinite, but precise, which is expressly opposed to any limit, and so clearly conceived that all its elements have their distinct and express properties.Does this mean that I perceive distinctly the infinite series of these elements? Certainly not. Here again, there is a substitute, the formula, from which the series and the properties of its elements are derived. What we perceive, is a general character of the dividend and remainder. After the first division we can see that, the remainder being 2 like the dividend, must, in becoming in its turn a dividend, give rise again to a remainder 2, which in its turn will do the same, and so on. In other words, we discover in the dividend this property of giving rise to a similar figure, which, being similar to it, has the same property as it. This abstract quality is the cause of the whole series; it forces it to be infinite, it alone is what we perceive; when we say that we conceive an infinite series, it only means that we discover this property of inexhaustible regeneration; all we seize is the generating law; we do not embrace all the engendered terms.-But as far as we are concerned the effect is the same; for by applying the law we are able to define whatever term we

please of the series, to measure exactly the increase of approximation it brings to the quotient, to calculate strictly the degree of error which the division would include were we to stop here. The perception of the law is equivalent to the perception of the series; an infinite line of distinct terms finds its substitute in an abstract character, and, in place of an experience which is by definition impossible, we have disengaged a property whose isolation has only required two experiences, and which is of equal value to us.

And so it happens, whenever we conceive and affirm some really infinite abstract magnitude, time, or space. We take a fragment, some short portion of the duration comprised in our successive sensations, some narrow portion of space comprised in our simultaneous sensations. We consider this fragment apart; we extract from it this property it has of being over-extended by a border absolutely similar to itself. We lay down, as before, a general law that the magnitude in question is continued beyond itself by another wholly similar magnitude, and this by another, and so on, without the possible intervention of a limit. Our conception of infinite time and infinite space is reduced to this.— But the result is the same as if the field of our imagination were infinitely extended and capable of setting before us at a glance the whole infinite successión we call time, or the extension, infinite in three directions, which we call space. For starting from the general character which alone is present to our minds, we are able to imagine any portion of time or space as clearly, and to affirm of it as surely, as if we had experience of it; no matter what the portion be, whether a fragment of duration preceding the solar system, or a portion of space beyond the furthest nebulæ of Herschel. It is possible, then, to represent infinite objects, series, or quantities,* by an abstract property. It is enough if this is their generator. By it, indirectly, they become present. Here we have, I think, the most

* When we speak of an infinite quantity, it is by extension; strictly speaking, a quantity is always finite, and there is nothing infinite but series.