behind. Weak and infirm as he is, he is obliged to undergo the fatigues of a long journey; hoping at the end of it to breathe in a land of liberty; he approaches his own country, flattering himself his reception there will confole him for his paft difgrace. But what am I going to fay? My heart finks, my hand trembles, and my pen falls to the ground: Let me be filent, therefore, on this affecting fubject.

"And, wherefore, am I thus treated? I do not fay for what reafon? but on what pretence?-The Magiftrates have been rafh enough to judge, me guilty of impiety, without reRecting, that the book, containing the pretended inftances of it,' is in the hands of the whole world. What would they not give, effectually to fupprefs this authentic teftimony against them, that they might be able more boldly to fay it contains what they pretend to have found there! But this proof of But this proof of my innocence will remain, in fpite of all their efforts to fupprefs it; and pofterity will be furprized, in looking for the enormous crimes imputed to the Author, to find at worst only, the errors and mistakes of a fincere friend to virtue."

Our ingenious and perfecuted Author goes on to mention the other aggravating circumftances of his oppreffion; hinting at Writers now living, who are more favourably dealt by, notwithstanding the principles they have inculcated in their works are notoriously fuch as he is unjustly accufed of: thefe, however, he forbears to name, as it is not his intention to injure others, but only to fhew the fingular hardfhip of his own cafe.

He obferves, that it is a ridiculous abfurdity for a Roman catholic Bishop, who condemns indifcriminately all that are not of his church, to cenfure any particular doctrine of a Proteftant Writer, as if he would not even permit thofe, whom he configns to the devil, to go to him their own way. He affects alfo to think it a mighty ridiculous thing, for fo many great States to enter into a league, as it were, againft fo mean an object as the fun of a Watch-maker. This reflection, however, we think ridiculous enough in our Author. We fhould have thought he had fuffered sufficiently, to be convinced of his own importance; which would not be a jot the lefs at prefent, had he been the fon of a Chimney-fweeper. A Writer, whofe works are become fo univerfal, and whofe opinions are fo well received as thofe of Mr. Rouleau, is, fingly, a man of more confequence, and may be more ufeful or pernicious to governments, than a score of Cardinals, or a whole junto of ordinary Minifters of State.

But the true caufe of our Author's perfecution in France, he hitelf conceives to be this. In a note, which was inferted in

his

1

his Eloifa, he had very unadvisedly spoken against the Janfenifts, predicting, that when they fhould get the upper hand, they would be more perfecuting than their enemies. ;-he had also refused to write against the Jefuits. At a time, when it was not yet determined to extirpate that fociety, this was overlooked, but not forgotten; those persons by whom the Parliament hath been excited to the present proceeding, having waited this opportunity of taking ample vengeance. On this account our Author rallies the Archbishop, on being fecretly made the dupe of that party, which he has had fortitude enough fo long to combat openly with fuccefs.

As a tranflation of this piece is advertised, and the prefent article already fufficiently long, we fall defer entering upon the merits of our Author's defence at prefent: this may probably be the fubject of a future article. In the mean time, we have only to obferve, that the whole of this epiftle is written with fpirit, and is worthy the pen of Mr. Rouleau.

The Nature and notable Ufe of the moft fimple Trigonal Numbers. With two Arithmetical Tables, that over and above the folution of feveral important Problems, give the Square-root out of every Square, expreffed by an integer Number, and feated between the Unity and forty thousand Millions; and the Cubic-root of every Cube, expreffed by an integer Number, and feated between the Intiger and two hundred fixteen thousand Millions. Tranflated

from the Latin of E. de Joncourt, A. M. and Profeffor of Philofophy; by the Author himself. Hague printed for Hufon. 4:0

HERE is fomething extremely agreeable in contemplating the Properties of Numbers. It opens an extenfive field, where the faculties of the human mind may range at large, in fearch of pleasure and utility. Many noble discoveries have been made, and many compendious methods of calculation invented, by this engaging ftudy. The common operations of Trigonometry, for instance, were extremely fatiguing till Lord Napier, by a happy difcovery of the property of numbers, formed the Logarithmic Tables, by which thefe operations are performed with the greateft facility.

THERE is fomething extremely a

The work, before us is an attempt of the fame kind, confift-, ing of tables of trigonal numbers, calculated principally to facilitate the extraction of the fquare and cube roots; the other arithmetical operations may be readily performed by these tables, which are eafily conftructed, in the following manner.

The

The natural numbers 1, 2, 3, 4, 5, 6, 7, 8, &c. being difpofed in a column, and the numbers 1, 3, 6, 10, 15, 21, 28, 36, &c. placed oppofite to them in a fecond column, a series of the moft fimple triangular or trigonal numbers will be generated, correfpondent to a feries of natural numbers, as in the following example.

The first trigonal number 1, is equal to the first natural number 1; the fecond trigonal 3, equal to the fum of the two first natural numbers 2+1=3; the third trigonal, 6, equal to the fum of the three firft natural numbers 1+2+3=6; &c. In the fame manner the large tables of trigonals, given us by this ingenious Author were conftructed.

From the very nature of the conftruction, it evidently appears, that the fum of any two trigonals following each other, is equal to the fquare of the natural number belonging to the larger trigonal: thus, for inftance, the fum of the trigonals 36 and 28, is 64, which is equal to the fquare of 8, (the natural number belonging to the greater trigonal 36) 8X8. Thus, by an eafy procefs of addition, the fquare of any whole number less than 20,000, may be very eafily found. But tho' the tables before us extend no farther than 20,000, yet the Author has fhewn, by an easy artifice, how any number, to 200,000, may be squared.

The rule which the Author has laid down for extracting the fquare-root is this: divide the given refolvend into two equal parts, and feek the half thereof among the trigonals; the number immediately above that half points to the natural number, or root required.Thus, if 49 were the given refolvend, the half will be 241, and the trigonal number immediately above, or greater than 24, is 28, and its natural number 7, the root required. But as most refolvends are furds, that is, have no true root, the Author has fhewn, by an eafy procefs, how the root may be approximated to any degree of accuracy.

To this work is annexed a table of the first 600 cubes, and their roots, by means of which the Author has fhewn how the

cube

cube-root of any number less than sixteen thousand millions may be readily extracted.

In fhort, M. Joncourt is the firft that has fhewn the practical ufe of thefe artificial numbers, and taken the pains to calculate tables for that purpofe; and therefore his work cannot fail of being agreeable to those who are pleased with seeing speculations reduced to practice.

We mention the practical use, because the doctrine of figurate numbers, (fo called from their being capable of reprefenting certain geometrical figures, by a particular difpofition of their units) is a part of the ancient Pythagorean fpeculations on numbers and geometrical figures; from a comparison of which, they pretended to difcover many myfteries and fecrets of nature. But fuch pretences have been long fince exploded, and the connections and properties of these numbers confidered as a fubject purely arithmetical; tho' they still retain their ancient

names.

They are all no other than the fums of different series of numbers in arithmetical progreffion; and are diftinguished by the common difference in the feries. Thus, if the common difference in the rank of progreffionals, whence they proceed, or whose fums they are, be an unit, as 1, 2, 3, 4, 5, &c. the fums 1, 3, 6, 10, 15, &c. are called triangles, or trigonal nuinbers. If the difference be 2, as 1, 3, 5, 7, 9, &c. the fums I, 4, 9, 16, 25, &c. are called quadrangles, and particularly fquares. If the difference be 3, as 1, 4, 7, 10, 13, &c. the fums, as 1, 5, 12, 22, 35, &c. are called pentagons, or pentagonal numbers; and fo on.

But it is not incumbent on us to purfue this fubject any farther; thofe who are defirous of feeing the doctrine fully explained, may confult Malcolm's New Syftem of Arithmetic, book V. page 396, where the connections and properties, of thefe numbers are difplayed in a masterly manner.

We are obliged to Mr. Joncourt for the compliment paid us, in dedicating his work to the Monthly Reviewers: but we wifh he had called in to his affiftance, fome friend better acquainted with the English language, which few foreigners write with any tolerable degree of elegance.

De Dea Libertate ejufque cultu apud Romanos et de Libertinorum Pileo Differtatio. Roma 1762. Or,

A Differtation on the Goddess Liberty, and the Worship paid

her

her among the Romans; as alfo on the Cap worn by the Freed-men of ancient Rome.

THIS is a very learned and ingenious enquiry, worthy of the elegant pen of the Abbé Venuti, its Author. As the fubjects of it, however, may be thought rather curious than important, by the generality of our Readers, we beg leave to refer the Antiquarian to the treatise itself.

Memoire fur l'Ufage Economique du Digefteur de Papin, &c. A Clermont-Ferrand. Or,

An Essay on the Economical Ufe of Papin's Digestor.

THIS is an account of an attempt to improve on this well

known machine; and, by making it cheap and commodious, to apply it to culinary ules. Mr. de Ballinvilliers, Intendant of Auvergne, hath occafioned many experiments of its utility to be made, in reducing the bones of animals into foup; which being rendered portable by evaporation, he thinks may be of public benefit to mankind, if diftributed, in times of scarcity, among the poor.

Eclairciffement fur les Moeurs, par l'Auteur des Mours. 12mo, Amfterdam, 1762. Or,

An Illustration of the Work intitled Manners. By the Author.

MR.

R. Touffaint, the celebrated Author of Les Mours, apologizes, in the prefent performance, for fome exceptionable paffages in that work; declaring, in the most pofitive terms, that whatever conftruction may have been put on fome unguard ed expreffions in his book, he looks upon the imputation of Deifm as the groffeft calumny; and that he then was, and now is, perfectly orthodox in his fentiments of Chriftianity. The publication of this apology, will probably be deemed much too late, to prevent the ill effects of the premature and inconfiderate fallies of his youthful genius. He endeavours to justify himself, however, in this delay; and, if his plea be not very folid, it is, at leaft, fpecious.

MONTHLY