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ditional history of Peru, any inftance of rebellion against the reigning prince, and, among twelve fuccellive monarchs, there was not one tyrant.

Even the wars in which the Incas engaged, were carried on with a fpirit very different from that of the other American nations. They fought not, like favages, to destroy, and exterminate; or, like the Mexicans, to glut blood-thirsty divinities with human facrifices. They conquered, in order to reclaim and civilize the vanquished, and to diffuse the knowledge of their own intitutions and arts. Prifoners feem not to have been exposed to the infults and tortures, which were their lot in every other part of the New World. The Incas took the people whom they fubdued under their protection, and admitted them to a participation of all the advantages enjoyed by their original fub jects. This practice, fo repugnant to American ferocity, and resemb ling the humanity of the most polished nations, must be afcribed, like other peculiarities which we have obferved in the Peruvian manners, to the genius of their religion. The Incas, confidering the homage paid to any object but the heavenly powers which they adored, as impious, were fond of gaining profelytes to their favourite fyftem. The idols of every conquered province were carried in triumpli to the great temple at Cuzco, and placed there as trophies of the fuperior power of the divinity who was the protector of the empire. The people were treated with lenity, and inftructed in the religious tenets of their new matters, that the conqueror might have the glory of having added to the number of the votaries of his father the Sun,

The ftate of property in Peru was no lefs fingular than that of religion, and contributed, likewife, towards giving a mild turn of character to the people. All the lands capable of cultivation were divided into three fhares. One was confecrated to the Sun, and whatever it produced was applied towards the erection of temples, and furnishing what is requifite towards celebrating the public rites of religion. The other belonged to the Inca, and was fet apart as the provifion made by the community for the fupport of government. The third and largest fhaic was referved for the maintenance of the people, among whom it was parcelled out. No perfon, however, had a right to ex clufive property in the portion allotted to him. He poffeffed it only for a year, at the expiration of which a new divifion was made, in proportion to the rank, the number, and exigencies of each family. All thofe lands were cultivated by the joint induftry of the community, The people fummoned by a proper officer, repaired in a body to the fields, and performed their common tafk, while fongs and mufical intruments cheered them to labour. By this fingular diftinction of territory, as well as by the mode of cultivating it, the idea of a common intereft, and of mutual fubferviency, was continually inculcated. Each individual felt his connection with those around him, and knew that he depended on their friendly aid for what increafe he was to reap. A ftate thus constituted may be confidered as one great family,

Herrera, dec. 5. lib. iv. c. 4. + Herrera, dec. 5. lib. iv. c. 8. Herrera, dec. 5. lib. iv. c. 2.

4

Vega, lib. v. c. 12.

Vega, lib. v. C. 5.

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in which the union of members was fo complete, and the exchange of good offices fo perceptible, as to create ftronger attachment, and to bind man to man in clofer intercourfe, than fubfifted under any form of fociety established in America. From this refulted gentle manners, and mild virtues unknown in the favage state, and with which the Mexicans were little acquainted."

Not that inequality of condition was unknown among the ancient Peruvians. On the contrary, the diftinction of ranks was fully established in Peru; among whom a great body of the inhabitants, under the denomination of Yanaconas, were held in a ftate of fervitude. Like the Tamemes of Mexico, they were employed in carrying burthens and in performing every other work of drudgery. The next above thefe, in rank, were their ordinary freemen, fuch as were diftinguished by no official or hereditary honours. Above them again were the Orejones, or Nobles, invested with offices of power or truft; and, at the head of all, the Children of the Sun; who, by their high defcent and peculiar privileges, were as much exalted above the Orejones, as thefe were elevated above the people.

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Our author proceeds to defcribe the state of the useful and elegant arts among the Peruvians; whofe unwarlike fpirit, he obferves, precipitated their fubduction by the Spaniards; who reduced them with all imaginable eafe; whereas the Mexicans maintained the struggle in defence of their liberties, with fuch perfevering fortitude, that it was with difficulty the Spaniards triumphed over them.-Perhaps, fays our author, the influence of those inftitutions, which rendered their manners gentle, gave their minds this unmanly foftnefs; perhaps the conftant ferenity and mildnefs of the climate may have enervated the vigour of their frame; perhaps fome principle in their govern, ment, unknown to us, is the occafion of this debility. What ever may have been the caufe, the fact is certain, and there is not an inftance in hiftory of any people fo little advanced in refinement, so totally deltitute of military talents and enterprize. This Character has defcended, continues Dr. Robertfon, to their pofterity; the Indians of Peru being now more tame and depreffed than any people of America.

Of the eighth book, containing a defcription of the Spanifh fyftem of Colonization and the prefent State of Spanish America, we shall speak in our next; concluding this Article.

S.

The

The Rationale of Circulating Numbers, with the Investigations of all the Rules and Peculiar Proceffes used in that Part of Decimal Arithmetic. To which are added, feveral curious Mathematical Queftions; with fome useful Remarks on Adfected Equations, and the Doctrine of Fluxions. Adapted to the Ufe of Schools. By H. Clarke. 8vo. 5s. Murray.

Although we are not fo fully perfuaded, as this writer feems to be, of the high regard, which ought to be paid, particularly in Schools, to the nature and utility of Infinite Circulating Decimals, they may have their ufe among the higher clafs of Arithmeticians at leaft it may not be improper for fuch to be made acquainted with the History, Rationale, and Manner, of working them.

"The first Specimen of Decimal Arithmetic that we meet with, is in the Aftronomical Tables of Arzachel, a Moor, who was very eminent in Spain about the beginning of the eleventh Century. They are adapted to the Meridian of Toledo; and as they are calculated for the Arabian Year of the Hegira, were probably originally written in Arabic: The Perfians, Moors, Arabs, and Saracens, being about that Period very famous for their Knowledge in Aftronomy. In these Tables, the Places of the Heavenly Bodies are denoted by a centefmal Divifion of the great Circles of the Sphere, to which the Arabian Algorithm of Numbers was better accommodated than the Greek or Roman literal Notation which had been hitherto made Ufe of for the Egyptian Sexagefms in the Aftronomical Tables of Ptolemy, Albateg nius, Abenazra, and other ancient Writers. Gerard Voffius informs us alfo of a Treatife entitled De Algorithmo, written by Johannes de Sacro Bofco, about the middle of the twelfth Century, who made Ufe of a centefmal Notation for the Extractions of the Square and Cube Roots. Aout the Year 1460, John Muller, fometimes named Regiomontanus, published his Book De Triangulis, in which he had conftructed a Table of Sines to the Radius 10,000,000; an Account of which may be feen in the Opus Palatinum de Triangulis, by Otho and Rheticus. The next Improvement in this Part of Arithmetic, we find in a Treatife entituled Arithmetica Memorativa, composed in Latin Verfe, by William Buckley, about the Year 1530, wherein he has given a Rule for extracting the Square Root of a Fraction; the Operation being nearly the fame with the prefent Mode of extracting the Square Root of a Surd Number, excepting that it is limited to a certain Number of Cyphers: The Rule, as corrected by Dr. Wallis, is,

Quadrato numero, fenas præfigito Cipbras:

Productii Quadri, Radix, per mille jecetur.
Integra dat Quotiens; & pars ita resta manebit,
Radici ut være nè pars milliffima de fit.

Referring to the Product of the Nume.ator and Denominator, mentioned in a former Kuls.

The

The Denominator being written under this Number, expreffes the Square Root of the Fraction. Afterwards Peter Ramus, in his Arithmetic, written about the Year 1570, and published by Schoner, fhews the Method of approximating to the Square and Cubic Roots of Surd Quantities, by adding Punctuations of Cyphers, exactly in the manner we now practice. But the first Treatife written profeffedly on this Subject, was published at Leyden, 1585, by Simon Stevens, entituled, DISME, or Decimals; which he tells us in his Geography, he believes to have been in Use among the Indians, and other Eaftern Nations, long before the Sexagefimal Notation was introduced by Ptolemy, in the Time of M. Aurelius. After this Time, Decimals began to be frequently ufed in Arithmetical Calculations, and were particularly much advanced by Briggs and Gellibrand, in their Trigonometria Britannica; by Oughtred, in his Clavis Mathematica denuò limata: alfo Wingate, Baker, Kerfey, and feveral other Authors of lefs Note, all contributed towards their Perfection, in their different Treatifes of Arithmetic. Yet we do not find, that any Regard had been paid to the Nature of Infinite Circulating Decimals before Dr. Wallis's Time. He was, in all probability, the first who distinctly confidered this curious Subject, as he himself informs us in his Treatife of Infinites. But he has neither given the demonftrations, nor fhewn their application. The latter of these defects, Mr. Brown, in his Decimal Arithmetic, and afterwards Mr. Cunn in his Treatife of Fractions, attempted to fupply, by giving Rules for their Operations. The former indeed has done this only in one fingle Cafe; but the latter has extended it to all Cafes. But as these are also wanting in the main Point, namely, a Demonftration, and are moreover defignedly expreffed in fuch a Manner, as to fet the Rationale of the Thing as far out of View as poffible; it is neceffary that either the Memory must be loaded with every Rule, or the Book be continually at Hand. Several other Authors have treated on Circulating Decimals. Martin, in his Decimal Arithmetic, has given fome practical Rules, but hath not sufficiently demonstrated them. Emerfon, in his Cyclomathefis, is excellent in the Theory, but has omitted the practical Part. Pardon, Vyfe, Thompson, and fome others, have alfo touched on this Subject; but as they all feem to have borrowed from Cunn, they are in the fame Predicament."

Malcolm and Donn, Mr. Clarke obferves, are the only authors who have treated the doctrine of Circulates in an intelligible manner; although to thefe he objects fome deficiencies, which he undertakes to fupply, as well as to retrench thofe fuperfluities, with which Cunn and others have loaded the theory of Circulates; the whole bufinefs, according to this writer, depending on, or to be deduced from, one fingle operation; viz. that of finding a finite vulgar fraction equivalent to an infinite repeating decimal.

Of the remaining contents of this volume Mr. Clarke gives the following account.

VOL. VI.

B

"As

"As the Operations of Circulates (as well as all other, Arithmetical Calculations) are most easily performed by Logarithms, I have fhewn the Method of finding the Logarithm of any Repeating Decimal; whereby the whole Bufinefs is greatly facilitated, and the Difficulty and Intricacy of the Rules by Common Arithmetic avoided. And, for the Amufement of fuch Pupils as have touched on the first Principles of Algebra and Geometry, I have inferted a few Questions, chiefly Originals, with their Solutions; and fome are given without Solutions, which are intended for the Exercife of thofe that are farther advanced. I have also added several Remarks on those Parts of the Mathematics which feem to the young Reader to be rather obfcure, namely, On Cardan's and Colfon's Theorems for Cubic Equations, wherein a very clear and concife Rule is given for extracting the Cubic Root of an impoffible Binomial; by which Cardan's Theorem is rendered generally useful, in finding the Roots of an Equation when they are all real, as well as when there is but one real and two imaginary-On the improbability of obtaining general Formula for the Surfolid and other higher Equations-On the Method of tabulating Literal Equations, illuftrated by Examples; from whence the Reverfion of a Series, however affected with Radicals, may be easily performed-On the direct and inverfe Method of Fluxions, wherein the Principles are fully explained, and by avoiding all Metaphyfical Confiderations, rendered clear to the loweft Capacity. The whole Bufinefs of finding Fluxions is reduced to one general Rule; and the particular Forms of fluxionary Expreffions are fo diftinguished, that the Learner may almost immediately determine in what Manner the Fluent may be obtained-On the Correction of a Fluent, and the Reafon of it-On Trigonometrical Fluxions, with their great Importance in Aftronomy-On the Phænomena of Saturn's Ring, being a new and curious Analytical Solution of the Problem refpecting the Times of its appearance and difappearance; whereby is alfo exhibited a new Species of Curves, &c. which is extracted from a Treatife juft published, entituled, Efai fur les Phénomènes relatifs aux disparitions périodiques de l'anneau de Saturne. By M. Dionis du Séjour, Fellow of the Royal Societies of London and Paris."

Of our author's method of illuftration, we fhall give a fpecimen from his Obfervations on the Nature of Fluxions, a fubject, whofe elucidation has been often attempted with little effect, on young perfons unaccustomed to metaphy fical fpeculation.

"The doctrine of prime and ultimate ratios, by which the fluxions of quantities are generally inveftigated, or demonftrated, contains in it fomething fo very obfcure and unintelligible to the learner, that it is rather more apt to confufe than give a proper arrangement to his ideas on the fubject. The most natural and eafy way of acquiring a right notion of fluxions, is by the introducing of time into the account. For by this means we do not confider them as mere velocities,

The first Lemma of Sir Ifaac Newton's Principia appears to many to be very exceptionable.

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