ART. V. La Langue des Calculs, &c. i. e. The Language of Calculation; a posthumous and elementary Work, printed according to the Author's original Manuscript; in which, certain Observations made on the Commencement and Progress of this Language expose the Defects of ordinary Language, and shew how in all Sciences the Art of Reasoning may be reduced to a well constructed Language. By CONDILLAC. 8vo. pp. 480. Paris. 1798. Imported by Dulau and Co. London. THE "HE author of this posthumous and unfinished work is known to the world by his Cours d'Etude, written for the instruction of the Duke of Parma. In the present treatise, he traces all computation to its origin; explains the causes which render ordinary language inadequate to the solution of questions, when at all complicated; and enforces the necessity of mathematical language, discussing its nature, its peculiar excellence, and the grounds on which its perfection is to be attempted. A work of this character merits attention; since, although the fundamental truths on which all science rests have easily gained universal reception, and pure geometry addressing the mind through the eye has been little subjected to çavil, yet the analytic art has furnished matter of much dispute and absurdity. The operations of algebra are mechanical; various and intricate combinations of quantities are produced; and many authors, not attentive to the circumstances under which they were obtained, have given either obscure, imperfect, or perverse explanations of the principles and methods of algebra. Certain properties have been assigned to quantities as inherent and essential, which depend solely on an arbitrary notation. The plain and obvious meanings of certain formulas have been neglected, to seek for latent truths or fanciful analogies. Hence, in many treatises, the science is obscure, perplexed, and mysterious. The student, in his first efforts, finds difficulties crowd fast upon him; he meets with definitions of which he discerns not the use, terms which oppress his memory, and refined and subtle reasonings which elude his comprehension. He therefore travels on slowly and wearily, cheered only now and then in his way by the faint light of a partial illustration. When mathematical reasoning is made to appear so distinct from all other reasoning, he fancies that the comprehension of it demands altogether a novel exertion of the understanding, and the calling forth of latent powers. Regarding what he reads as truths above the reach of controversy, he is obliged to believe that to which he cannot assent; his memory is forced into activity, but his judgment becomes inert i mert; he loses the spirit of inquiry and examination, and lulls Such is not unexceptionably the character of all treatises; Before the time of CONDILLAC, writers had told the world that algebra was merely a language; that numeration was the real basis of all computation; that multiplication, involution, &c. were only compendious methods of addition; and that the pure and abstract sciences were composed of series of identical propositions These important truths, however, it is the merit of the Abbé CONDILLAC to have fully explained and confirmed. We do not say that he has placed every subject of discussion entirely beyond doubt and dispute: but he never endeavours to mislead his readers into the wilds of fanciful conjecture, and the mazes of subtle refinement; he never clothes obscure and imperfect notions in pompous and mysterious language; and if he sometimes quits the straight road, he almost immediately returns to it.. We shall now give some extracts from the work: Chap 4. In what consist the ideas of numbers? The sciences are great and beautiful paths, traced and opened by nature, the entrances to which have been closed by men: they have aukwardly placed there briars and obstacles of all kinds: they have even hollowed out precipices; so that at present the whole difficulty consists in taking the first steps. We see, in the attempts which have been made to effect a passage, only the confused traces of uncertain wanderers. Some men of genius, indeed, find their way but they are in a certain degree removed from our sight, and they refuse to inform us how they discover the passage, or they purposely conceal it. As we do not then aistinctly conceive how they conquered the obstacles, we imagine that they bounded over them; and we figure in our minds these men floating on the air, while we find ourselves doomed to take step after step along the earth :-but do we more clearly perceive the means by which they bounded over the obstacles or soared above them? certainly not; let us try then to clear the entrance that has If this enterprize has been closed; there is no other passage for us. difficulties, they are not so great as they may at first appear to be.. Besides, when they are surmounted, we shall find ourselves in those beautiful paths, in which men of genius have travelled before us; and they will probably confess that their progress was like ours, step by step, and along the earth. My sole concern, in my outset, is to rid myself of every thing This is the reason why at first I move slowly; that embarrasses me. this is the reason why I stop so long at questions which calculators never never once deemed it necessary to discuss; because these questions belong to metaphysics, and calculators are not metaphysicians. They are ignorant that algebra is only a language; that this language is morcover as yet without a grammar, and that metaphysics alone can furnish it with one. We have seen that, for every finger which we open, we pass by numeration to a number greater by unity; and by denumeration to a number less by unity, for every finger that we close. • When we are accustomed to represent by our fingers a series of numbers alternately decreasing and increasing, we perceive that we are able to represent the same series by all other objects, by stones, by trees, by men, &c.; that is to say, we perceive ourselves able to perform numeration and denumeration by stones, trees, men, &c. in the same manner as by our fingers. • The ideas which we have formed by means of our fingers we apply by analogy to stones, trees, men, &c. and since we can apply them to all objects in the universe, we say that they are general, that is, applicable to every thing: but, when we merely consider them as applicable to every thing, we do not apply them to any particular objects; we consider them by themselves, and separate them from all things to which they can be applied. Nevertheless, it is in the very objects that we originally perceived these ideas, and there only have we been able to perceive them. At first we saw them in our fingers, as we remarked the order in which they successively opened and shut. Next we saw them in all other objects, as we could count by their means in the same manner as with our fingers. To consider numbers, then, in a general manner, or as applicable to all objects in the universe, is the same thing as not to apply them to any of these objects in particular; it is the same thing as to abstract or to separate them from these objects, in order to consider them apart; and then we say that the general ideas of numbers are abstract ideas. When the ideas of numbers, at first perceived in the fingers, and afterward in all objects, become generalized and abstracted, we perceive them no longer in the fingers, nor in the objects to which we cease to apply them. Where, then, do we perceive them? In the names which are become the signs of numbers. There remain in the mind only these names, and we search in vain for any thing else there. One, two, three, &c. here then are the abstract ideas of numbers: for these words represent numbers as applicable to every thing, and as applied to nothing. These words separate the numbers from the objects in which we have learnt to perceive them. When, for example, having said one finger, one stone, one tree, we say one without expressing any object, we then have in that word, one, abstract unity. If you think that abstract ideas are any thing else than mere names, say, if you can, what they are? In fact, when you have made abstraction of fingers, and other objects which can represent numbers: when you have made abstraction of names, which are other signs of numbers; you will in vain search your mind for something that remains, there is nothing, absolutely nothing. • But But, it will be said, how can abstract ideas be so reduced as to be only words? It will be more easy to me to answer this question, than it would be to answer the following :-If abstract ideas be not mere words, what are they? • Numbers are represented to me by the fingers, when I learn to perform numeration; and they are represented to me by other objects, when I repeat with them what I have learnt with the fingers. As I represent them, I give to each a different name: I designate by one a finger considered alone; and consequently I shall say one of a stone, or of a tree: I express by two one finger added to one finger; and consequently I shall say two of a stone added to a stone, or of a tree added to a tree.. I shall act similarly with the names three, four, &c. But what ideas do these names recall? I answer that one is a word which I recollect to have chosen to signify a single finger, a single stone, a single tree, and in general an individual object; that two is a word which I recollect to have chosen to represent a finger added to a finger, a stone added to a stone, a tree added to a tree, and in general an individual added to an individual. As in general names, however, such as one, two, three, there exist properly only names: so also in abstract ideas there are properly only names: for abstract ideas and general names are in reality the same thing. • The error committed on this subject proceeds from a supposition that the word idea has only one acceptation. It has two : one which is peculiar to it, and another which is given to it by extension. If I say one stone, two stones, the word idea is taken in its proper sense, for I find the ideas of one and of two in the objects which I join to these names: but if I say one, two, these are only general names; and it is by extension only that they can be called ideas. It is known that except among us there is neither genus nor species: it is known that there are only individuals, although our philosophers, who without doubt are aware of this truth, forget it so often that they appear to be ignorant of it. Genus and species, then, are only denominations of our own invention; and this was needful, since the confinement of our understandings imposed on us the necessity of classing objects. But the denominations given to numbers are only methods of classing things, in order to observe them under the different relations in which they appear during calculation: for the same reason, then, that in the universe there is no such thing as genus and species, there is also no such thing as two, three, four; in a word, no such thing as a number; there are only, if I may so express myself, one, one, one; and numbers are only in the names which we have made for our use. In the eye of God, there is no number: as he sees every thing at one glance, he counts not. We are obliged to count, because we see things one by one only; and, in order to count, we are obliged to say, two, three, four, as if there really was something which was two, three, four. We even suppose it ;-naturally inclined to realise our abstractions, we willingly establish this principle that every thing, which we clearly and distinctly conceive, is, independently of us, such as we conceive it to be. A good Cartesian will not doubt it.' The The Abbé CONDILLAC, in his essay " De l'Art de Raisonner," (p. 12.) had said that to demonstrate is to change the terms of a definition, and to arrive by a succession of identical propositions at a conclusion identical with the proposition from which it is immediately deduced. This notion is dilated and made sensible by instances in the present treatise; and on this subject we shall quote a passage: It will be said, if, in every subject that is studied, we proceed from property to property by a series of identical propositions; each property, as each proposition, is in this series the same as that which precedes it, and by consequence all are reduced to one and the same property. How then are they one many? How are the first, the second, the third, to be distinguished? Although a property be one, it may be considered under many points of view; and it would be one to us as it really is, if we could comprehend all the points of view at once. This we are unable to do; and hence it is that we first consider it in one relation, then in a second, and so on successively; that to us it becomes a first property, ́a second, a third, &c. We must not, then, imagine that such consequences are in things themselves, they are only in our language; and every science may be reduced to a primary truth, which, transformed from one identical proposition to another, offers to us, in a series of transformations, all the discoveries which have been made, and all which remain to be made. True it is that, in order thus to seize the sciences, we must speak with the greatest simplicity: for it is our badly constructed language that opposes the greatest obstacles to the progress of knowlege. We should know how to invent, if we knew how to speak; but we speak before we have learnt, and we do not love simplicity. I therefore plainly foresee that the method followed by me in this chapter will not be generally approved. What! it will be exclaimed, must we, in order to acquire knowlege, drag ourselves heavily along from identical propositions to identical propositions? I answer that it is necessary so to do, and that inventors dragged themselves along as we must: if we doubt it, 'tis because, when they shew us their discoveries, they are on their feet, and suffer us to believe that they have always been so. They are not less superior geniuses, however, if they arrived at their discoveries by trailing along the ground: this only proves that they were men, and that the human mind is very limited let us conclude that, whatever our acquirements in know. lege may be, we have nothing to make us vain,-nothing even about which we can be affectedly modest: the true philosopher is neither the one nor the other.' Having remarked that the words hundred, ten, &c. are general signs, and by extension only called general ideas; and that all operations in calculation are mechanical, with whatever signs performed; the author adds, It may be hence inferred, perhaps, and objected against me, that the general ideas of metaphysics are not properly ideas; that they are only |