Died: 1643 in Paris, France

**Pierre Hérigone** is actually a pseudonym for the Baron Clément Cyriaque de Mangin. In fact, just to make things even more confusing, Cyriaque de Mangin also used the pseudonym Denis Henrion. He was of Basque origin. Little is known of his life except that he taught for most of it in Paris.

Hérigone's only important work is the six volume *Cursus mathematicus, nova, brevi, et clara methodo demonstratus* or, to give it its French title, *Cours mathematique, demonstre d'une nouvelle, briefve, et claire methode* which appeared between 1634 and 1642. In fact remaining copies of the first five volumes were reissued with new title pages in 1644 when the sixth volume was published. This is in fact a fictitious second edition, since it consisted of rebound copies of the first edition volumes which had been unsold. The work is a compendium of elementary mathematics written in both French and Latin. It introduced a complete system of mathematical and logical notation, yet none is used today. Florian Cajori wrote:-

A full recognition of the importance of notation and an almost reckless eagerness to introduce an exhaustive set of symbols is exhibited in the Cursus mathematicus of Pierre Hérigone, in six volumes, in Latin and French ,...

Hérigone did, however, introduce a number of familiar symbols such as for 'is perpendicular to'. He also used < for 'angle' but since this is essentially the same as the 'less than' symbol, William Oughtred made a slight modification to Hérigone's notation using ∠ for 'angle' in *Trigonometria* (1657). At a time when an exponent notation was not in common use, Hérigone introduced *a*2, *a*3, etc for the powers of *a*. This is a variant of our present notation in which the powers are raised while in the form in which Hérigone used them they were on the same level as the *a*.

To give a little flavour of this rather strange work, let us note that Hérigone describes a camera obscura in the form of a goblet which was designed so that one could spy on others at the table while drinking from the goblet. It had a cunning optical set-up with a mirror but appears only to have been created for a bit of fun. He also introduced a code by which numbers were translated into words to aid memorising them. The code was as follows: 1 = *p*, a; 2 = *b*, e; 3 = *c*, i; 4 = *d*, o; 5 = *t*, u; 6 = *f*, *ar*, ra; 7 = *g*, *er*, re; 8 = *l*, *ir*, ri; 9 = *m*, *or*, ro; 0 = *n*, *ur*, *ru*. So to remember a number such as 314159 one produced a word such as 'cadator' which then translated back into 314159. The assumption here was that 'cadator' was easier to remember than 314159.

One of the problems which Hérigone looked at in the text concerns an apothecary who had four kinds of medicines, of which the first is hot in the fourth degree, the second is hot in the second degree, the third is cold in the first degree and the fourth is cold in the third degree. He then asks in what proportions the medicines have to be mixed to produce a compound of the first degree of heat. He solves it in a rather clumsy way using a geometrical proof with unnecessary use of several theorems from Euclid's *Elements.* There are also some gems in the text such as one described in the article [2] where a clear description of the determination of the sine law by Hérigone in the fifth volume of his *Cursus mathematicus* appears.

We know that Hérigone served on a number of committees and took a full part in the mathematical life of Paris. One committee which he served on was set up to judge whether Jean-Baptiste Morin's scheme for determining longitude from the Moon's motion was practical. The committee members, in addition to Hérigone, were Etienne Pascal, Mydorge, Beaugrand, J C Boulenger and L de la Porte. Hérigone, and the rest of the committee, became involved in a dispute with Jean-Baptiste Morin.

We mentioned above that Hérigone also wrote under the name of Denis Henrion. It was under this pseudonym that he translated *Tractatus de Globis* by the English geographer Robert Hues (1554-1632). The original text was written in Latin and published in London in 1594. The translation into French by Denis Henrion was published as *Traite des Globes et de Leur Usage, traduit du Latin de Robert Hues, et Augmente de plusieurs nottes et operations du Compas de proportion par D Henrion, mathematicien* in Paris in 1618. As can be seen from the title, Hérigone added his own notes on the use of the compass of proportion. Finally we mention *Quatre livres de la géométrie pratique* published by Cyriaque de Mangin in 1620.

**Article by:** *J J O'Connor* and *E F Robertson*

**August 2006**

[http://www-history.mcs.st-andrews.ac.uk/Biographies/Herigone.html]