Universal arithmetic : this is how M. Newton refers to algebra, or the calculation of magnitudes in general;  [1] and it is not without reason that this name has been given to it by this great man, whose spirit, equally luminous and profound, seems to have penetrated all sciences to their true metaphysical principles. In fact, in ordinary arithmetic , one can note two kinds of principles: first there are the general rules, independent of the specific signs by which one expresses the numbers; the others are the rules that depend upon these same signs, and it is these that are called more specifically the rules of Arithmetic . But the first sets of principles are nothing other than the general properties of relationships, that in some way occur so that the relationships are denoted; such, for example, are these rules, that if one subtracts one number from another, this other number, when added to the remainder, will yield the first number; if one divides one quantity by another, the quotient multiplied by the divisor will yield the number that was to be divided; if one multiplies the sum of several numbers by the sum of several others, the product is equal to the sum of the products of each multiplied by the other, etc.

From this it follows first of all that in denoting numbers by general expressions, that is, ones that do not denote one number any more than another, one will be able to form certain rules related to the operations that one can perform on the numbers so denoted. These rules come down to representing, in the simplest possible manner, the result of one or of several operations that can be undertaken on numbers expressed in a general manner; and this result thus expressed will properly be only an indicated arithmetical operation, an operation that will vary accordingly when one gives different arithmetical values to the quantities, which, in the result to which they refer, represent the numbers.

The better to understand this notion that we give of algebra, let us run through the four ordinary rules, and let us begin with addition. This consists, as we have seen in the article Addition, of the adding together with their signs, without any other operation, of dissimilar quantities, and the addition of the coefficients of similar quantities: for example, if I am to add together the two dissimilar quantities a, b , I will write simply a+b ; this result is only another manner of indicating that if one sets a for one number, and b for another, the two numbers are to be joined together; thus a+b is only the indication of an arithmetical operation, the result of which will be different according to the numerical values one assigns to a and to b . For the present I assume that if someone proposes to me to add 5 a to 3 a , I could write 5 a +3 a , and the arithmetical operation would be indicated as above: but in examining 5 a and 3 a , I see that this operation can be indicated in a simpler way: for whatever number a represents, it is evident that this number multiplied five times, added to the same number multiplied three times, is equal to the same number multiplied eight times; thus, I see that in place of 5 a +3 a , I can write 8 a , which is a briefer expression, and which indicates to me an arithmetical operation that is simpler than is indicated to me by the expression 5 a +3 a .

The above is the foundation of the general rule for algebraic addition, to sum up similar quantities by adding their numerical coefficients, and thereafter writing the part represented by letters once.

One can then see that algebraic addition can be reduced to expressing, in the simplest manner, the sum or the result of several generally expressed numbers, and, so to speak, leave the arithmetician with only the least work it is possible to do. It is the same with algebraic subtraction; if I wish to deduct b from a , I simply write a-b , because I cannot represent this in any manner that is simpler; but if I have to deduct 3 a from 5 a , I will never write 5 a -3 a , because that would give me several arithmetical operations to do, in case I should wish to give a a numerical value; I will write simply 2 a , a simpler and more convenient expression for arithmetical calculations. See Subtraction.

I would say the same for multiplication and division: if I wish to multiply a+b by c+d , I could just as well write either ( a+b ) x ( c+d ), or ac+bc+ad+bd , and often even I would prefer the first expression to the second, as it appears to demand fewer arithmetical operations; for the first requires but two additions and one multiplication, while the second requires three additions and four multiplications. But if I have to multiply 5 a by 3 a , I will write 15 aa instead of 5 a x 3 a , because in the first case, I would have three arithmetical operations to perform, while in the second I have only two, one to find aa , and the other to multiply aa by 15. Likewise, if I have a+b to multiply by a-b , I will write aa-bb , because this result is often more convenient than the other for arithmetical calculations, and hence I derive from this a theorem, which is that the product of the sum of two numbers multiplied by the difference of these two numbers, is equal to the difference between the squares of these two numbers. [2] It is thus that we find that the product of a+b multiplied by a+b , that is, the square of a+b , would be aa+2ab+bb , [3] and that consequently it contains the squares of the two numbers, plus twice the product of one number multiplied by the other; this is used to extract the square roots of the numbers. See Square and Square root. [4]

As for division, in place of writing
, I will simply write 4 a ; in place of writing
, I will write a-x . But if I have to divide bc by hd , I will write
, it not being possible to find a simpler expression.

We see then from this that M. Newton was right to call algebra universal arithmetic ; as the rules of this science consist merely of extracting, so to speak, what there is that is general and common to all the particular arithmetics that one can perform with more or less or as many numbers as ours, and to present in the simplest and briefest way these indicated arithmetical operations.

But, one will say, what is the good of all this piling up of proofs? In all the questions one could propose about numbers, each number is designated and stated. What is the use of giving that number a letter value, which it seems one can do without? Here is the advantage of this naming.

None the questions one can propose for numbers are as simple as to add one given number to another, or to subtract them, or to multiply or divide one by another. This is about much more complicated questions, for whose solutions one must make combinations, into which the number or numbers one seeks must enter. It is therefore necessary to know the art of making these combinations without knowing the numbers one seeks, and to do that one must express numbers by characters that differ from numerical characters, because it would be very inconvenient to express an unknown number by a numerical character that might not match it except by the greatest chance. To make this easier to understand by an example, let us suppose we are searching for two numbers whose sum is 100, and whose difference is 40: I see first of all that in designating these two unknown numbers by numerical characters chosen at will, for example the first by 25 and the second by 50, I would give a very false result, as 25 and 60 [5] do not at all satisfy the two conditions thus expressed: x plus y is equal to 100, and x minus y is equal to 60; [6] or in algebraic characters:

As x + y is equal to 100, and x-y equal to 60, I note that 100, combined with 60, has to be equal to x + y , combined with x-y . Hence to add x + y to x-y , we must, following the rules of algebraic addition, write 2 x ; I then see that 2 x is equal to 160, that is to say that 160 is double the largest number being sought; therefore this number is half of 160, that is 80: from there it is easy to find the other that is y : for as x + y is equal to 100, and as x is equal to 80, thus 80 plus y is equal to 100; therefore y is equal to 100 from which one has taken 80, that is 20; hence the two numbers being sought are 80 and 20: their sum is 100, and their difference is 40. [7]

Furthermore, I do not claim to have this article reveal the need for algebra; for it would not be needed at all, if one did not propose questions more complicated than those here; by this very simple example, obvious to everyone, I just wanted to show how, with the help of algebra, one can find unknown numbers.

The algebraic expression of a question is nothing other than, as M. Newton put it so very well, the translation of this same question into algebraic notation: a translation that is so easy and basic, that it is reduced to what is absolutely necessary for the question, and that all superfluous elements have been eliminated. We follow M. Newton in providing the following example.

Thus the question comes down to finding three unknowns, x, y, z by the three equations xz=yy, x+y+z =20, xx+yy+zz =140. All that remains is to derive from these three equations the value of each of the unknowns.

Hence we see that in universal arithmetic there are two parts to be distinguished.

The first is that which teaches how to make combinations and the calculation of quantities represented by signs that are more universal that numbers; in such a manner that the unknown quantities, that is to say those whose numerical value is unknown, can be combined with the same ease as known quantities, that is to say those to which one can assign numerical values. These operations establish only the general properties of the quantity, that is to say that here one envisages the quantity just as a quantity, and not as one represented and fixed by this or that particular expression.

The second part of universal arithmetic consists of knowing how to make use of the general method for calculating quantities, in order to discover the quantities one seeks by means of the quantities one knows. For that one must 1°. represent in the simplest and most convenient manner the ordering of the relationship that there must be between the known and the unknown quantities. This ordering of the relationship is what is called an equation ; thus the first step to take, when one has a problem to solve, is first of all to reduce the problem to the simplest equation.

Then one must derive from this equation the value or the different values that the unknown that one is seeking should have; this is what is called solving the equation . See the article Equation, where one will find greater details, to which we refer, having in this article to confine ourselves to giving a general idea of universal arithmetic , in order to detail its rules in particular articles. See also Problem , Root, etc.

The first part of universal arithmetic is properly called Algebra or the science of calculating magnitudes in general, while the second is properly called Analysis : but these two names are used often enough for one another. See Algebra and Analysis.

We do not know whether the ancients knew this science: however, it appears that they had some similar method by which to resolve numerical questions; for example, the questions that have been called the questions of Diophantus . See Diophantus; [8] see also Application of Analysis to Geometry .

According to Abbé de Gua, [9] in his excellent history of Algebra, of which one will find the major part in the article Algebra in this Dictionary, Theon appears to have believed that Plato was the inventor of Analysis, and Pappus informs us that Diophantus and other ancient authors – such as Euclid, Appolonius, Aristeus, Eratosthenes and Pappus himself – primarily applied themselves to it.  [10] But we do not know precisely what their analysis consisted of, and in what it either differed from, or resembled ours. M. de Malezieu, in his Elémens de Géométrie , [11] contends that it is morally impossible that Archimedes was able to come to the greater part of his wonderful geometrical discoveries, without the help of something equivalent to our Analysis: but all of this is only a conjecture; and it would be most unusual if there did not remain at least some vestige of this in some of the writings of the ancient geometers. M. de l’Hopital, or even more M. de Fontenelle, who is the author of the preface of the infinitely small , [12] observes that it appears that M. Pascal came, by force of reasoning alone and without use of Analysis, to the wonderful discoveries that make up his Traité de la roulette , published under the name d’ Etonville .  [13] Why would it not have been the same for Archimedes and the ancients?

We have so far spoken of the use of algebra only for the resolution of numerical questions: but what we have just said about analysis among the ancients, leads us naturally to speak of the use of algebra in geometry: this usage consists mainly of solving geometrical problems by algebra, as one would resolve numerical problems, that is, by giving algebraic terms to known and unknown lines, and after having expressed the problem in algebraic terms, by calculating in the same manner as if one were to solve a numerical problem. What we call in algebra an equation of a curve , is nothing but an indeterminate geometrical problem, for which all the points on a curve give the solution: and thus the rest. In the application of algebra to geometry, lines that are known or given are represented by letters of the alphabet, like the known or given numbers in numerical questions: but it must be noted that the letters that represent lines in the solution of a geometrical problem cannot always be expressed by numbers. Suppose, for example, that in the solution of a problem of geometry, one has two known lines, of which the one that I shall call a is the side of a square, and the other that I shall name b is the diagonal of the same square; I say that if one assigns a numerical value to a , it will be impossible to assign a numerical value to b , because the diagonal of a square and its side are incommensurable. See Incommensurable , Diagonal , Hypotenuse, etc. Thus, algebraic calculations applied to geometry have an advantage, in that the characters that express given lines can mark quantities that are commensurable or incommensurable. It is true that the unknown number being sought can be represented by an algebraic expression that denotes an incommensurable: but then this is an indication that this number that is unknown and being sought, does not exist at all, and that the question cannot be resolved except in general terms and not exactly; whereas in the application of algebra to geometry, one can always assign, by a geometrical construction, the exact magnitude of the unknown line, even if the expression that denotes that line is incommensurable. One can often even assign the value of this line, even though one cannot express it by an algebraic expression, whether it be commensurable, or incommensurable: this is what happens in an irreducible case of the third degree. See Irreducible Case.

One of the greatest benefits that one has taken from the application of algebra to geometry is differential calculus; one will find more on the subject at the word Differential, together with an exact notion of calculus. See Calculus and Integral.

There is today no geometrician, no matter how little capable, who does not know more or less the infinite utility of these two calculations in transcendental geometry.

M. Newton has given us an excellent work on algebra, which he entitled Arithmetica universalis . In this he lays out the rules of the science, and its application to geometry. In it he gives several new methods, which for the most part have been commented upon by M. s’Gravesande in a very useful little work for beginners, entitled Elementa algebroe ,  [14] and by M. Clairault in his elements of algebra.  [15] See in the article Algebra the names of several other authors who have examined this science; we believe that the work of M. s’Gravesande, that of Father Lamy, [16] the science du calcul by Father Reyneau, the Analyse démontrée by the same author,  [17] and the Algebra of Saunderson published in English, [18] are in this genre the works from which young men could profit the most, while in many of these treatises, and perhaps in all, there is a wealth of desirable things. On the manner of applying algebra to geometry, that is to say, the reduction of geometrical questions to equations, we know of nothing either better or more luminous than the rules given by M. Newton, page 82 ff of his Arithmetica universalis , in the Leiden edition of 1732, up to page 96 . These are too precious to be abridged, and too long to be inserted here in their entirety, thus we send our readers there. We say only that they can be reduced to two rules.

First rule . A geometrical problem having been proposed (and one could say as much of a numerical problem), compare together the known and unknown quantities that bound the problem; and without distinguishing the known from the unknown, examine how all these quantities depend on one another; and which are those that being known will cause the others to be known, in proceeding by a synthetic method.

Second rule. Among the quantities that would lead to knowing the others, and that for this reason I call synthetics, search for those that would lead to knowing the others the most easily, and which could be the most difficult to find, if one supposed them to be completely unknown; and regard these quantities as those you should treat as known.

It is upon this that is founded the rule for geometers, who say that in order to resolve a geometrical problem algebraically, one must suppose it already solved: in effect, to solve this problem, one must imagine all the lines, both known and unknown, as quantities one has before one’s eyes, and which all depend upon one another; so that the known and the unknown can be treated reciprocally and in their turn, if one wants, as unknown and known. But enough of this matter in a work in which one should only lay out general principles. See Application .

1. Isaac Newton’s Cambridge lecture notes on algebra were collated and published by William Whiston, Newton’s successor as the Lucasian Professor of Mathematics at the University of Cambridge, as Arithmetica universalis : sive de compositione et resolutione arithmetica liber (London: Benj. and Sam. Tooke, 1707). The linked volume is the second edition, published in 1722.

2. What d’Alembert means here is that “... is equal to the result of the first squared number being divided by the second squared number.” This can most easily be demonstrated by the simple example, (5+3)x(5-3)=16, as the sum of the first parenthesis is 8, and the sum of the second parenthesis is 2; the square of the first number is 8x8=64, while the square of the second number is 2x2=4; and the result of the division of the first squared number (64) by the second squared number (4) is (64/4=16), which is also the result of the original operation.

3. That is, using the same numbers as before, [(5+3) x (5+3)=64] is equivalent to [(5x5)+2(5x3)+(3x3) = 64].

4. Although there is no article on the square root, the article “Root of a number” discusses both square roots and cube roots. (ed.)

5. [ Sic ] in the original; should be “50”, based on context.

6. [ Sic ] in the original; should be “40”, based on context.

7. This entire calculation is wrong, as using 80 and 20 for x and y , respectively, does yield a sum of 100 for x + y , as stated, but a difference of 60 , not “40” as stated in the text for the original conditions, for x-y . To satisfy the conditions x + y =100 and x-y =40 (as originally specified) one can use only x =70 and y =30.

8. Diophantus of Alexandria was a third-century CE mathematician whose Arithmetica examined the solution of problems by equations. The link is to the Latin translation by Gulielmus [Wilhelm] Xylander, published in Basel in 1575.

9. Jean Paul de Gua de Malves (1713-1786) was a French mathematician and early member of the Encyclopédie project. His Usages de l'analyse de Descartes pour découvrir, sans le secours du calcul différentiel, les propriétés... des lignes géométriques was published in Paris in 1740.

10. Theon of Alexandria and Pappus of Alexandria were both 4 ^th -century CE mathematicians.

11. The first edition of Elemens de Géométrie de Monseigneur le Duc de Bourgogne ... by Nicolas de Malezieu (1650-1727) was published in Paris in 1705; link is to the third, posthumous edition of 1729.

12. The marquis de l’Hospital (1661-1704) published his Analyse des infiniment petits, pour l'intelligence des lignes courbes in Paris in 1716.

13. Blaise Pascal’s notes on roulette were included in his pseudonymous Lettres de A. Dettonville contenant quelques-uns de ses inventions de geometrie, (Paris, 1659).

14. Willem Jacob ’s Gravesande (1688-1742) was a Dutch mathematician who helped to diffuse Newton’s work across continental Europe.

15. Alexis Clairault (1713-1765) was a French mathematician who developed Newtonian methods and, in 1736, participated in the geodesic expedition to Lapland led by Pierre Louis Maupertuis. Clairault’s Elemens d’Algebre was published in Paris in 1746.

16. Father Bernard Lamy (1640-1715) was a French theologian and scientist whose principal mathematical work was in mechanics and equilibria. His work Elemens des mathematiques, ou Traite de la grandeur en general ... was first published in Amsterdam in 1680.

17. Charles-René Reynaud (1656-1728) was a French mathematician whose work summarized trends in mathematics in the early part of the 18 ^th century. La science du calcul des grandeurs en general, ou, Les elemens des mathematiques was first published in Paris in 1714-1736 ; Analyse demontrée, ou La méthode de résoudre les problêmes des mathematiques, et d’apprendre facilement ces sciences was first published in Paris in 1708.

18. Nicholas Saunderson (1682-1739) was Lucasian Professor of Mathematics at the University Cambridge (1711-1739). The Elements of Algebra in Ten Books was published posthumously by Cambridge University Press in 1740. Diderot famously used him as an example in his Letter on the Blind for those who Can See , first published in 1749 and available in English translation in Diderot’s Early Philosophical Works, trans. Margaret Jourdain (Chicago, 1916).