Histoire de l'Académie des Sciences de Paris. Année MDCCIII. Avec les memoires de mathematique & de physique, pour la même année (Paris: Boudot, 1705)
pp 58-63

Date: 1703

Note: This essay was Fontenelle's introduction to Leibniz's "Explanation of Binary Arithmetic" when it was published in l'Histoire de l'Académie des Sciences de Paris in 1703.

Translated from the French

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[Hist p58]

New Binary Arithmetic

     The science of numbers is so natural to men, cultivated for so many centuries by so many excellent minds, and as of now is so thoroughly perfected, that [Hist p59] a new arithmetic, completely different from the one we follow, must be a kind of wonder.
     Yet, to consider the matter more closely, as the foundation of all our arithmetic is purely arbitrary, it is permissible to take another foundation, which will give us another arithmetic. It was desired that the first and fundamental sequence of numbers be an infinite succession of tens. But it is evident that to have extended the fundamental sequence of numbers to ten, or not to have extended it further, it is an institution which could have been different. And it even seems that it was done by chance by the common folk, and that mathematicians were not consulted, for they would have readily established something more convenient. For example, if the sequence of numbers had been pushed up to twelve, the thirds and quarters which are not in ten would have been found there without fraction.
     Numbers have two kinds of properties, some essential, the others dependent on an arbitrary institution and the manner of expressing them. That the odd numbers always added in succession give the natural sequence of square numbers is an essential property in the infinite series of numbers, however it is expressed. But that in all the multiples of 9, the digits which express them added together always make 9, or a multiple of 9 lower than the one proposed, it is a property which is by no means essential to the number 9, and it has it only because it is the penultimate number of the tenfold progression it pleased us to choose. If the progression to twelve had been taken, 11 would have had the same property.
     It is very convenient to be able to recognize at first glance, and without [performing] any operation, that 25245, for example, is a multiple of 9, and if mathematicians had established the fundamental progression which should reign in arithmetic, they would, after [Hist p60] having examined them all, have preferred the one that would have produced the most similar conveniences, either for common and popular use, or for scholarly research.
     Mr. Leibniz, having studied the simplest and shortest of all the possible progressions, which is that which ends in two, found it very rich and very abundant in these kinds of accidental properties. There would be in the whole of his arithmetic only two digits, 1 and 0. The zero would have the power to multiply everything by two, just as in ordinary arithmetic it multiplies everything by ten. 1 would be one, 10 two, 11 three, 100 four, 101 five, 110 six, 111 seven, 1000 eight, 1001 nine, 1010 ten etc., which is entirely based on the same principles as the expressions of the common arithmetic.
     In truth, this would be very inconvenient because of the large quantity of digits it would need, even for very small numbers. For example, [in binary arithmetic] it takes 4 digits to express eight, which we express with one. So Mr. Leibniz does not want to get his arithmetic into popular use; he claims only that for difficult research it will have advantages that the other does not, and that it will lead to higher speculations.
     It was in 1702 that he communicated to the Academy this binary arithmetic,1 announcing only that it would have great uses for the sciences but not revealing them. He did not want it to be mentioned in the Histoire [de l'Académie Royale des Sciences] until this new invention could be accompanied by its uses.
     In the present year, it was found that it had one, which Mr. Leibniz himself had not expected. Jesuit Father Bouvet, renowned Missionary in China, to whom Mr. Leibniz had written about the idea of his binary arithmetic, told him that he was quite convinced that this was the true meaning of an ancient Chinese enigma, [Hist p61] bequeathed over 4000 years ago by Emperor Fuxi, founder of the sciences and empire of China, and apparently understood in his century and several centuries after him, but of which he [Bouvet] was certain that understanding had been lost for more than 1000 years, in spite of the research and efforts of the most learned scholars, who had only drawn puerile and chimerical allegories. This enigma consists in the different combinations of an whole line and a broken line, repeated a certain number of times, either one or the other. Assuming that the whole line means 1, and the broken line means 0, we find the same expressions of numbers as binary arithmetic. The conformity of the combinations of the two lines of Fuxi, and of the two unique digits of Mr. Leibniz's arithmetic, struck Father Bouvet, and made him think that Fuxi and Mr. Leibniz had had the same thought. If the truth of this happy encounter is confirmed, what glory for Europeans, at least in the eyes of the Chinese, to have given them the key to their ancient science: it is always certain that by thinking as much as we are doing now, and by turning around in so many different ways a certain matter, and a certain stock of rational thoughts given to men, it is impossible that we do not find almost everything that other centuries have thought of better.
     If Mr. Leibniz did not have the same thoughts on binary arithmetic as Emperor Fuxi, at least Mr. de Lagny did have the same thoughts as Mr. Leibniz on this very subject.2 As we have already seen in the Histoire of 1702, Mr. de Lagny, Professor of Hydrography at Rochefort, works to perfect the science he practices. In relation to navigation, he undertook a new trigonometry, and when studying the whole system of logarithms, which were invented mainly for trigonometry, he saw flaws and inconveniences therein, for which he could not find a remedy other than by imagining binary arithmetic.

[Hist p62]

     The great convenience of logarithms is to change multiplications and divisions, which are long and difficult operations for large numbers, into additions or subtractions, which are much simpler and easier. But Mr. de Lagny claims that this advantage that theory promises so magnificently reduces to nothing in practice, that on the contrary, as logarithms, which are kinds of feigned and supposed numbers, are a circuitous route one takes to arrive at natural numbers, the only ones sought, there is always further to go, although perhaps more easily, and always a longer time required to use, and he calls as his witness all those who have calculated by this method. He even argues that logarithms are false in large numbers, and gives as proof a calculation that Henry Briggs in his Arithmetica logarithmica, p27f, gave as an example of the use of logarithms.3
     In binary arithmetic, multiplications and divisions are necessarily done by simple additions and subtractions, without having to go through any circuitous route, such as that of logarithms in common arithmetic, and consequently all the advantage that common arithmetic draws from logarithms only forcibly is essential to binary arithmetic, the multiplications and divisions of which Mr. de Lagny for this reason calls natural logarithms.
     He set out his idea more fully in a writing he printed this year at Rochefort,4 and which he sent to the Academy, but the little we have said will be enough to put on the right track those who will want to go further into this new arithmetic.
     As the greatest mathematicians can very legitimately be jealous of the glory of having had the same thoughts as Mr. Leibniz, without having followed him, we owe this testimony here to Mr. de Lagny, that having always been at Rochefort, he does not appear to have had any [Hist p63] knowledge of what Mr. Leibniz had sent to the Academy on the binary calculus.


1. Leibniz in fact communicated it in 1701, in his "Essay on a new science of numbers" (26 February 1701).
2. Thomas Fantet de Lagny (1660-1734).
3. Henry Briggs, Arithmetica logarithmica (Gouda: Rammasensius, 1628).
4. Fontenelle is not consistent on this matter. In a separate essay, he identifies the work as Trignonmétrie française ou reformée and indicates that it is not yet published: "It was said above that Mr. de Lagny is working on a new trigonometry. He will call it Trigonometrie Françoise or Reformée, a title which responds in part to that of Trigonometria Britannica by Briggs. In this new trigonometry, Mr. de Lagny puts in place of the old logarithms, which he finds arbitrary and defective, the natural logarithms of binary arithmetic. He also has new views on the tables of sines, tangents, and secants, and he has given to the Academy a small sample of his work on tangents and secantes and an assurance of his promises." Bernard le Bovier de Fontenelle, "Sur les tangentes et les secantes des angles," in Histoire de l'Académie Royale des Sciences. Année MDCCIII. Avec les memoires de mathematique & de physique, pour la même année (Paris: Boudot, 1705), 64-65. If such a work by de Lagny was ever written, it was never published and is no longer extant. It was certainly planned, because de Lagny himself referred to it in an essay published in 1725; there he proposes "to divide the quadrant into 30 degrees, one degree into 32 minutes, one minute into 32 seconds etc." and states that "this is the subject of a preliminary dissertation of my Trigonometrie Françoise or Reformée." Thomas Fantet de Lagny, "Second memoire sur la goniometrie purement analytique, ou methode nouvelle & generale, pour déterminer exactement, lorsqu'il est possible, ou indéfiniment près lorsque l'exactitude est impossible, la valeur des trois angles de tout triangle rectiligne, soit rectangle, soit obliquangue, dont les trois côtés sont donnés en nombre; & cela par le seul calcul analytique sans tables des sinus, tangentes & secantes," in Histoire de l'Académie Royale des Sciences. Année M.DCCXXV. Avec les memoires de mathematique & de physique, pour la même année (Paris: L'imprimerie royale, 1725), 282-323. However, no evidence of de Lagny's work on binary arithmetic, if there ever was any, has survived. The only source for the claim that de Lagny worked on the binary system is Fontenelle.

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