Arithmetic is intimately connected with astronomy. A complete decimal numeration was known to all the civilized American nations, and even to the northern tribes. The most rude tribes reckoned by 5 or the manual mode; there are traces also of a binary numeration, the most simple of all: while others had complex calculations by 13 and by 20 or scores. We find no trace of any by 7, and but slight indications of a ternary numeration by 3 and 9. All these American modes of reckoning may thus be reduced to the _binary_, by two or pairs; and the quinary or manual by the five fingers, of which the decimal is the double, and by 20 the quadruple. 8. _Languages._ They are becoming one of the most important aids in history. When the annals are ample and clear, the examination of the languages is merely a supplement to historical knowledge; but when they are obscure, mutilated or totally lost; languages then supply more or less to their defects or loss. Their comparative study furnish us new lights to ascertain the origines, parentage, dispersions, colonies, alliances, wars, &c., of the nations thus deprived of written annals or even traditions. They serve also to rectify the imperfect annals or the fabulous traditions. This study may lead besides to trace the manners, religions, intercourse, arts and sciences of nearly all nations; since the proper languages of each people offers a picture of the civilization, acquired or borrowed knowledge, modes of life, &c. of each. It is above all in both Americas that this study is indispensable in historical researches: I will therefore apply to languages in all cases, and make constant use of them; and they will unfold new facts quite unknown, although very important, Historical lights shall thereby be thrown on many obscure subjects, whence astonishing and unexpected results may spring, in which I shall depend as much as upon mutilated and neglectful traditions. By taking into view all the American languages or as many as are already known, we shall easily dispel the errors and absurd systems of philosophers and philologists, who taking only a few as samples of the whole, have either deemed _all the Americans_, as many Jews, or Tartars, or Atlantes, or sprung from the ground, and so on. Now the fact is that these writers have never taken the trouble to compare the numerous American languages and dialects, reduce them to groups, and seek their affinities elsewhere. Adelung and Vater had once stated without proof, that nearly 1200 languages existed in America. Balbi has reduced them to 423, of which 212 in South America; but they can be much further reduced, most of them being mere dialects. The whole may be comprised in 25 groups of languages, or even less; which were certainly identic in 25 languages 2 or 3000 years ago: and all of which have astonishing affinities with the groups of the eastern hemisphere, so as to indicate a parentage 4 or 5000 years ago.