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

Even the wars in which the Incas engageıl, 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 inititutions and arts. Prisoners feem not to have been exposed to the insults and tortures, which were their lot in every other part of the New World. The Incas took the people whom they subdued under their protection, and admitted them to a participation of all the advantages enjoyed by their original tub. 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 observed in the Peruvian manners, to the genius of their religion. The Incas, considering the homage paid to any object but the heavenly powers which they adored, as impious, were fond of gaining proselytes to their favouriie fyítem. The idols of every conquered province were carried in triumpli to the great temple at Cuzco *, and placed there as trophies of the superior power of the divinity who was the protector of the empire. The people were treaged with lenity, and initructed in the religious tenets of their new malters , that the conqueror might have the glory of having added to the number of the votaries of his father the Sun,

"" The state of property in Peru was no leis singular than that of religion, and contributed, likewile, towards giving a mild turn of caracter to the people. All the lands capable of cultivation were divided into three shares. One was consecrated to the Sun, and whatever is produced was applied towards the erection of temples, and fornishing what is requisite towards celebrating the public rites of religion, The other belonged to the Inca, and was ser apart as the provision made by the community for the support of government. The third and largest Mare was reserved for the maintenance of the people, among whom it was parcelled out. No perfon, however, had a right to ex clusive property in the portion allotted to him. He poflefled it only for a year, at the expiration of which a new division was made, in pro portion to the rank, the number, and exigencies of each family. All ihose lands were cultivated by the joint industry of the community, The people fummoned by a proper officer, repaired in a body to the fields, and performed their common talk, while songs and musical instruments cheered them to labour 1. By this fingular distinction of territory, as well as by the mode of cultivating it, the idea of a common interest, and of mutual fubferviency, was continually inculcated. Lach individual felt his connection with those around him, and knew that lie depended on their friendly aid for what increase he was to reap. A late thius conitituted may be considered as one great family,

* Herrera, dec. s. lii. iv. c. 4. Vega, lib. v. C. 12.
+ Herrera, dec. 5. lib. iv. C. 8.
1 Herrera, dec. 5. lib. iv. c. 2. Vega, lib. V. c. :5.

in which the union of members was so complete, and the exchange of good offices fo perceptible, as to create stronger attachment, and to bind man to man in clofer intercourse, than fublifted under any form of fociety established in America. From this resulted 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 distinction of ranks was fully established in Peru; among whoin a great body of the inhabitants, under the denomination of Yanaconas, were beld in a state of servitude. Like the Tamemes of Mexico, they were employed in carrying burthens and in performing every other work of drudgery. The next above these, in rank, were their ordinary freeinen, such as were distinguished by no oficial or hereditary, honours. Above them again were the Orejones, or Nobles, invested with offices of power or trust; and, at the head of all, the Children of the Sun; who their high descent and peculiar privileges, were as much exalted above the Orejones," as these were elevated above the people.

Our author proceeds to describe the state of the useful and elegant arts among the Peruvians ; whose unwarlike spirit, he oblerves, precipitated their subduction by the Spaniards; who reduced them with all imaginable ease; whereas the Mexicans maintained the struggle in defence of their liberties, with such persevering fortitude, that it was with difficulty the Spaniards triumphed over them.-Perhaps, says our author, the influence of those institutions, which rendered their manners gentle, gave their minds this unmanly softness; perhaps the constant Serenity and mildness of the climate may have enervated the vigour of their frame ; perhaps some principle in their govern, ment, unknown to us, is the occasion of this debility. What, ever may have been the cause, the fact is certain, and there is not an instance in history of any people so little advanced in refinement, fo totally dettitute of military talents and enterprize. This Character has descended, continues Dr. Robert, Son, to their pofterity; the Indians of Peru being now more tame and depressed than any people of America,

Of the eighth book, containing a description of the Spanith system of Colonization and the present State of Spanish America, we shall speak in our next; concluding this Article.

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The Rationale of Circulating Numbers, with the Inveftigations

of all the Rules and Peculiar Processes used in that Part of Decimal Arithmetic. To which are added, several curious Mathematical Questions ; with some useful Remarks on Adfelled Equations, and tbe Doctrine of Fluxions. Adapted to the Use of Schools. By H. Clarke. 8vo. 55. Murray.

Although we are not fo fully persuaded, 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 use among the higher class of Arithmeticians : at leaft it may not be improper for such to be made acquainted with the History, Rationale, and Mannery of working them.

The first Specimen of Decimal Arithmetic that we meet with, is in the Astronomical 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 s and as they are calculated for the Arabian Year of the Hegira, were probably originally written in Arabic: The Persians, Moors, Arabs, and Saracens, being about that Period very famous for their knowledge in Astronomy. In these Tables, the Places of the Heavenly Bodies are denoted by a centesmal 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 Use of for the Egyptian Sexagesms in the Aftronomical Tables of Ptolemy, Albategnius, Abenazra, and other ancient Writers. Gerard Voslius informs us also of a Treatise entitled De Algorithmo, written by Johannes de Sacro Bosco, about the middle of the twelfth Century, who made Use of a centesmal Notaiion for the Extractions of the Square and Cube Roots. . Aout the Year 1460, John Muller, sometimes named Regiomontanus, published his Book Þe Triangulis, in which he had constructed a Table of Sines to the Radius 10,000,000; an Account of which may be seen in the Opus Palatinum de Triangulis, by Otho and Rheticus. The next Improvement in this Part of Arithmetic, we find in a Treatise entituled Arithmetica Memorativa, composed in Latin Verse, 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 present Mode of extracting the Square Root of a Surd Number, excepting that it is limited to a certain Nunber of Cyphers: The Rule, as corrected by Dr. Wallis,

Quadrato numero *, senas prefigito Cipbras:
Productii Quadri, Radix, per mille fecetur.
Integra dat Quotiens; & pars ita rečia manebit,

Radici ut være pars millefima de fita * Referring to the Product of the Nume, ator and Denominator, mentioned in a former Ruls.

The

The Denominator being written under this Number, expresses the Square Root of the Fraction. Afterwards Peter Ramus, in his Arithmetic, written about the Year 1570, and published by Sehoner, hews 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 Treatise written professedly 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 Eastern Nations, long before the Sexagesimal Notation was introduced by Ptolemy, in the Time of M. Aurelius. After this Time, Decimals began to be frequently used in Arithmetical Calculations, and were particularly much advanced by Briggs and Gellibrand, in their Trigonometria Britannica ; by Oughtred, in his Clavis Mathematicæ denuò limata : also Wingate, Baker, Kersey, and several other Authors of less Note, all contributed towards their Perfection, in their different Treatisés 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 considered this curious Subject, as he himself informs us in his Treatise of Infinites. But he has neither given the demonstrations, nor fhewn their application. The latter of these defects, Mr. Brown, in his Decimal Arithmetic, and afterwards Mr. Cunn in his Treatise of Fractions, attempted to supply, by giving Rules for their Operations. The former indeed has done this only in one single Case; but the latter has extended it to all Cases. But as these are also wanting in the main Point, namely, a Demonstration, and are moreover designedly expressed in such a Manner, as to set the Rationale of the Thing as far out of View as possible; it is necessary that either the Me. mory 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 Arithinetic, has given fome praštical Rules, but bath not fufficiently demonstrated them. Emerson, in his Cyclomathelis, is excellent in the Theory, but has omitted the practical Part. Pardon, Vyse, Thompson, and some others, have also touched on this Subject; but as they all seem to have borrowed from Cunn, they are in the same Predicament."

Malcolm and Donn, Mr. Clarke observes, are the only authors who have treated the doctrine of Circulates in an ins telligible manner; although to these he objects fome deficiencies, which he undertakes to supply, as well as to retrench those fuperfluities, with which Cunn and others have loaded the theory of Circulates ; the whole business, according to this writer, depending on, or to be deduced from, one single 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

66 As

“ As the Operations of Circulates (as well as all other, Arithme. tical Calculations) are most easily performed by Logarithms, I have shewn the Method of finding the Logarithm of any Repeating Deçimal; whereby the whole Business is greatly facilitated, and the Difficulty and intricacy of the Rules by Common Arithmetic avoided. And, for the Amufęment of such Pupils as have touched on the first Principles of Algebra and Geometry, I have inserted a few Queitions, chiefly Originals, with their Solutions; and some are given without Solutions, which are intended for the Exercile of those that are farther advanced. I have also added several Remarks on those Parts of the Mathematics which seem to the young Reader to be rather obfcure, namely, On Cardan's and Colson's Theorems for Cubic Equations, wherein a very clear and concise 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 Formulæ for the Surfolid and other higher Equations-On the Method of tabu. lating Literal Equations, illustrated by Examples; from whence the Reverfion of a Series, however affected with Radicals, may be easily performed on the direct and inverse Method of Fluxions, wherein the Principles are fully explained, and by avoiding all Metaphysical Confiderations, rendered clear to the lowest Capacity. The whole Business of finding Fluxions is reduced to one general Rule; and the particular Forms of Auxionary Expressions are so distinguished, that the Learner may almost immediately determine in what Manner the Fluent may be obtained on the Correction of a Fluent, and the Reason of it-On Trigonometrical Fluxions, with their great Importance in Astronomy-n the Phænomena of Saturn's Ring, being a new and curious Analytical Solution of the Problem respecting the Times of its appearance and disappearance ; whereby is also exhibited a new Species of Curves, &c. which is extracted from a Treatise juft published, entituled, Esai 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 illustration, we shall give a specimen from his Observations on the Nature of Fluxions, a subject, whose elucidation has been often attempted with little effect, on young persons unaccustomed to metaphy, fical speculation.

“ The doctrine of prime and ultimate ratios, by which the fluxions of quantities are generally investigated, or demonitrated, contains in it something so very obícure and unintelligible to the learner, that it is rather more apt to confuse than give a proper arrangement to his ideas on the subject *. The most natural and easy way of acquiring a right notion of fluxions, is by the introducing of time into the ac. count. For by this means we do not consider thein as mere velocities,

The first Lemma of Sir Isaac Newton's Principia appears to many to ise yery exceptionable.

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