which is also absolutely incorrect. And suppose it were possible to avoid this example by some means or other, at what should we arrive?-clearly, at last, at what we desire; viz. at a general idea of "multiplication," of which, according to the views hitherto handed down, we have at present a total lack. But cannot similar questions be proposed respecting Powers and Logarithms ?

The fourth question: how may the paradoxes of calculation be most securely avoided?-obliges us to submit to a very exact examination the subject of mathematical analysis, its first and simplest ideas, as also the methods of reasoning which are applied in it. This is what the author of these sheets zealously did from 1811 to 1821, and he published the commencement of his investigations in 1816, and only six years afterwards, a very happy result of them in the first edition (1822) of the two first volumes of his "Attempt at a perfectly consequential System of Mathematics."

And although this last-mentioned work was not received without approbation,—although its sale has enabled him to publish five following volumes of the same work,-although his shorter Instruction-Books are already spread abroad in two or more editions,-yet he can very well recollect the time when on his first appearance he hardly escaped being declared "insane" by several mathematicians on account of his views; when he was even designated in official papers as a dangerous innovator on account of these revolutionary ideas of science, and most persons contented themselves either with a silent shrug of the shoulders, or with publicly accusing him of "presumption." This public resistance only drove the author to test and retest his views continually, and if possible more rigorously, whereby his works have received a better finish, and are better adapted to satisfy the demands of scientific unity, and yet he has been unable to discover that his (certainly revolutionary) fundamental view of the subject admitted of or required any essential alteration, since by it, all formerly observed contradictions are most harmoniously solved, or, more properly speaking, are not encountered, and can only appear as solved or avoided, when this new process is compared with the older one, which the author terms "that hitherto employed."

The author is at this moment convinced that he has only to let his Instruction-Books work on quietly and peaceably, in order to see his views adopted by most teachers, because those books are also distinguished (precisely on account of their predominant scientific unity) by great simplicity and didactical convenience. But inasmuch as professed mathematicians, as e. g. Abel was, do not generally occupy themselves with reading elementary instruction-books,—even when these would dry up the sources of their complaints, the author endeavours in these pages to exhibit his views to such persons, in as short and comprehensive a manner as possible, and at the same time to point out the most important

conclusions respecting a well assured and necessarily correct method of working with infinite series.

Now the author is convinced that all the difficulties which are met with in mathematical analysis, are to be attributed wholly and solely to the very first fundamental view, that view namely which each has taken for himself of the subject and nature of mathematical analysis. It appears to him namely as if the object had been confounded with the means which must be applied in order to attain that object. The object of war is-peace; but how unsuitable would it be, and to what erroneous consequences would it lead, if we declared too generally: war is the doctrine of peace," or, "war is occupied with what is peaceful"! Thus the object of mathematical analysis is perhaps in all cases nothing more than the comparison of magnitudes, but it is totally repugnant to the views of the author to say: "Mathematics," (and therefore mathematical analysis as a portion of the same,) "is the doctrine of magnitudes (quantities)." On the contrary,

the author has found himself forced to conceive the nature of mathematical analysis much more abstractly, and he believes that he is much nearer the truth in asserting that: "mathematical analysis is the doctrine of the relation of those (seven) (mental) acts to one another, to which we are led by the consideration of (whole, indenominate) number," i. e. therefore "the doctrine of the oppositions" and relations (combinations) in which the above named mental operations stand to (with) one another*.

b

Viewed from this point of sight, the forms a+b, a−b, a.b, α a', a, log a do not represent magnitudes (quantities), but mental acts (in systematic language: "symbolized operations"), which stand in certain relations to one another, that are enunciated in " 'equations." Every "equation" is a so-called identical one, and every such equation never expresses anything but the relation of the operations to one another, so that every new equation expresses the same relation, only in a new modification t. This is not only true for every equation between letters, but also for every equation between cyphers, as e. g. for the equation 65+24=89, only that in this equation the signification of the several cyphers has been attended to, so that it no longer appears in its original purity, as it would do if written thus,

65+24=(60+20) + (5+4).

*This is curiously enough the oldest as well as the newest view, the Arabic phrase from which our word Algebra is derived being "el-gebr wa-l-mukabalah,” which may be aptly rendered "combination and opposition.' Trans.

+ So called algebraical and transcendental equations (which the author has designated determining equations) can only receive meaning and authority as identical equations; and they do not differ at all from identical equations in their nature, but only in so much that one or more of their letters has to represent determinate (generally, unknown) expressions, which must be mentally substituted for these letters.

If we now consider these equations in their simplest form, as

a-(b-c)=a-b+c;

a-b a b

·; (a + b) c = ac+bc; &c. &c.

C

с C

66

we have the simplest laws which govern the operations (i. e. these mental acts); and the application of these laws to the formation of new and more complex equations" constitutes calculation," in which idea all and every kind of calculation, the commonest as well as the most advanced, is included.

66

There are only (indenominate) whole numbers; whatever else appears in practical calculation under the name of number or magnitude (quantity) (negative, broken, imaginary) is nothing but a combination of two or more whole numbers with one another, made by means of the above named mental operations, i.e. by means of symbolized operations. But if we start originally from whole indenominate numbers, the difference a b may be either reduced to a whole number or, it remains per se, and in this last case it gives the idea b-b or zero (0), or the idea 0-(b-a) or 0-c or c, (since it is usual to understand zero as the minuend of the difference, and not to write it down). We may also conceive the form 0+c, which is usually written + C. But +c and c and 0 are far from representing "magnitudes" (quantities); on the contrary, they only express the existence of combinations of numbers (i. e. of mental acts), which follow determinate laws, so that we are precisely for this reason enabled to calculate with these expressions.

a

In the same way the quotient or

α

-B

γ -0

may be reduced to a positive or negative whole number or zero, or, it remains per se; in the latter case arise the ideas of broken number, and of positive and negative broken number. These broken numbers are therefore according to this view not "magnitudes" (quantities), but expressions which announce the existence of mental operations that proceed according to determinate laws, so that we are enabled to calculate with these expressions.

The root a also allows of being reduced to a prior form (i.e. to a positive or negative whole or broken number or zero), or, it remains per se. But in the last case it always admits of being reduced to the simplest root per se, viz. to√-1, so that all expressions which contain roots per se receive the form p+q√−1. Finally the logarithm log a (in especial) never remains per se, but may be always reduced to an actual number (i. e. to a positive or negative whole or broken number or zero), or at any rate to an imaginary number, i. e. to the form p+q.-1.

b

According to this view the so CALLED ACTUAL NUMBERS are as far from being "magnitudes" as are THE IMAGINARY. The actual and imaginary numbers stand here in the same category; they are both of them nothing but forms per se, i. e. symbolized operations,

i. e. conceived and therefore real combinations of numbers effected by means of the above mentioned mental acts, i. e. expressions which embody these mental acts, while these last are governed by determinate laws which are enunciated in equations, so that these laws can be applied, i. e. so that we can " CALCULATE" with these expressions.

After having pointed out these general views we have now to shew that these abstract ideas possess a firm supporter. First of all it must be remembered that according to this view_the form of an expression is at the same time its essence. Form lost, all lost. Hence the form of the expression is the supporter of the idea, and the properties, which this form is obliged to possess, are the characteristics of the idea. Finally, the relation of the operations to one another is enunciated in the determination of what new form is to be substituted for an old, given form; hence the whole of mathematical analysis is solely employed in the transformation of given forms. Consequently it is not magnitudes (quantities), but forms which are the subject of mathematical analysis.

Hence we explain a sum to be an expression of the form a+b or a+b+c, &c. &c. endowed with the property, that its elements a and b, or a, b and c, &c. may be interchanged at pleasure, i. e. that ba may be written for a+b, or the new forms a+c+b, b + a +c, &c. instead of the old form a+b+c, without our having to expect any contradiction on the part of the laws of operation, (i.e. the laws of the mental acts which are here considered.)

Hence we explain a difference to be an expression of the form a-b, endowed with the property that (a − b) + b may be interchanged with a itself.

Hence we explain a product to be an expression of the form a.b or a.b.c, &c. &c. endowed with a double property, first that its elements a, b, c, &c. &c. may be interchanged at pleasure, and secondly that (a+b). c may be interchanged with ac + bc,* without our having to fear a contradiction on the part of the laws of operation, (i. e. of the mental operations when abstracted from whole numbers).

Hence we explain a quotient to be an expression of the form

a

endowed with the property that 7.6 may be interchanged

with a itself.

In the same way the power at must be explained to be an expression of this determinate form, and representing either other expressions with determinate properties, or these properties themselves per se.

Then we explain the root to be an expressson of the form /a, endowed with the property that (a) may be interchanged with a itself.

*The second property enunciates the connection between the product and the sum. The first property is common to both product and sum.

b

Finally, we explain the logarithm to be an expression of the form log a endowed with the property that bloga may be interchanged with a itself.

After establishing these most general ideas of the sum a+b,

the difference a-b, the product a. b, the quotient

b

a

,

the power ㄜˊ

a', the root Ja, and the logarithm log a, the ideas of "addition," "subtraction," "multiplication," "division," "potentiation," (or involution), "radication" (or evolution), and logarithmation" (of a to, from, by b) allow of being perfectly generally established as signifying respectively "the operations by means of which these 7 forms are constructed." Objectively considered therefore these operations consist in simply writing down these symbols a+b, a-b, &c. &c.

But since these ideas must not be allowed to receive contradictory characteristics, we must be particularly careful with respect to the ideas of product and power, that the double or triple properties with which we endow them are not contradictory. Now in as much as at the time when the idea of product is established, there only occur (since we originally started from indenominate whole numbers,) the whole number or the difference per se a-ẞ of two (such indenominate, whole) numbers, we can only at first establish the particular ideas of the product ab, (1) when a and b are whole numbers, and (2) when a and b are differences of whole numbers, and then prove that for these particular products no contradiction is to be expected. But at the time that the idea of power is established, we have already the (reducible and also the) quotient per se α-β whose dividend and divisor are such differences of whole numbers, i. e. in other words, we have the so-called actual number, with which we must not stand in contradiction.

7-8'

γ

According to this view the most difficult point is to settle the idea of "equation;" for although we have already definitely enough enunciated the object of the equation, yet if in accordance with that object we were able to define it thus: "two expressions, i. e. two forms (symbolized operations) are equal to one another, when they may be unconditionally substituted for one another, without our having to fear a contradiction on the part of the laws of operation," it would still be necessary to possess well defined external characteristics by which the correctness or incorrectness of an equation might be ascertained. Hence we see in reference to "equation," as we saw a little while ago in reference to "product" and "power" the necessity arise of arranging these views, which have been here preliminarily collected generally together, systematically, so that they may form a well connected whole which will furnish the requisite conviction of correctness in every case. Now if we