• with 4 common dice, we muft from 1146 fubtract the fum ⚫ of the chances for miffing all the aces, and throwing one precisely, that is, 1125, wherefore the chances required are 21,

Let there be a heap of 16 counters, whereof 6 are red, and 10 black ones, what is the probability, that in drawing two of them blindfold, they fhall be both red ones?

SOLUTION.

• Suppose them taken one by one. If the firft drawn should be a red one, of which the probability is the probability

6

16'

6 5

30 240'

• pening X (Art. vii. *) or the probability fought. 16 15

If the agreement be fuch, that either the firft or fecond counter drawn happening to be red, the drawer should be • entitled to a certain fum S, his expectation may be found • thus, If the first drawn counter is a red one, his expectation upon the second will entirely vanish, because drawing one red ⚫ counter infures to him the proposed fum, but if a black one ⚫ should be drawn firft, of which the probability is his ex

15

10

16'

× S, or x S. The

X

probability of fucceeding may be determined with more facility by finding the probability of drawing two black counters fucceffively, and fubtracting it from unity, the remainder being certainly the probability of drawing either both red C counters or one of each colour, but whichfoever of these shall happen, the drawer wins. Now the probability of drawing • two black counters fucceffively is

10

9

16X

15'

which fubtracted

The probability of two events (independent of each other) happening, is equal to the product of the probabilities, whereby these events may happen fingly.

⚫ from

15

from unity leaves

or

24' 8

5, this multiplied by S, gives the

value of the drawer's expectation, the fame as before.

PROBLEM V.

Let there be a lottery confifting of 100 tickets, wherein ⚫ there are four prizes; to find the probability that, in the three first tickets that are drawn, there fhall be one prize at least.

SOLUTION.

Here it is evident, that if we find the probability of the three first drawn tickets being all blanks, and fubtract it from unity, the remainder will be the probability of drawing one prize at leaft. The probability of drawing a blank the first • time is If a blank fhould be fo drawn, then there remains

96

100

95 blanks, and the probability of drawing the next ticket a

95 99'

• blank is that of the third ticket being a blank whence

96 95 94
X X

100 99 98'

94 981

is (Cor. Art. vii. *) the probability of

the three first drawn tickets being blanks, and therefore

96 95 94

I

-X X

or

941

100 99 98 of 8085'

is the required probability,

or that of taking one or more prizes in the three first drawn • tickets.

COROLLARY.

Hence it appears, that in many cafes it will be much more convenient to inveftigate the probability of the contrary to what is required in the problem to happen, rather than the probability of what is fpecified therein.'

This extract will be fufficient to fhew in how eafy a manner Mr. Clark has folved the problems in his introduction. But the reader must not expect that the more intricate problems in the work itself are done with the fame facility; yet, in juftice to the Author, we muft obferve, that they are, in general, folved in as eafy and confpicuous a manner as the fubject will admit.

Since the probability of the happening of two or three events may be confidered as compounded of the refpective probabilities of their happening feparately, it follows, that the probability of the happening of any number of events, i. e. that they fhall all happen, is equal to the product of the probabilities of thofe events happening confidered fingly.

B.

Account

Account of FOREIGN BOOKS.

Traité des Eaux Minerales de Spa, par Jean Philippe de Limbourg, Docteur en Medicine. Seconde Edition, revue, corrigée, & augmentée par l'Auteur, à laquelle on a joint une Carte des environs de Spa. That is,

A Treatife on the Mineral Waters of Spa, by Dr. Limbourg. The fecond edition, revifed, corrected, and augmented by the Author; to which is now annexed, a map of the country about Spa. Liege, printed for F. J. Defoer, 8vo.

THE

name.

HE Author opens his work with a preliminary discourse on the excellency of the fprings of Spa; of which he conceives it to be no flight proof, that in other countries, in order to recommend their mineral waters, they bestow upon them this He will, however, by no means allow, that in reality any of thofe mineral waters are comparable to these. He gives alfo, in this introduction, a lift of the feveral works that have been publifhed upon the fubject, as well thofe he has not feen, as thofe which he has confulted; and he affures us, the most antient author that he has met with, is Dr. Gilbert Lymborth, Phyfician and Canon of Liege, who, in 1559, obliged the Public with a differtation on the waters of Spa, printed at Antwerp. As the analysis of this work will afford the English Reader, in a narrow compafs, a curious, entertaining, and useful account of these celebrated fprings, we hope the following extract of our Author's laboured performance, will merit his approbation.

Spa is a town fituated in the marquifate of Franchimont, in the country of Liege, at the diftance of about eighteen miles. fouth-east from that capital. It lies in the midst of forests; and the little plains in its neighbourhood, are terminated on every fide by high mountains. The country round it, is far from being fruitful. It is not without difficulty, that they obtain edible roots, oats, and hay, and these but in very fmall quantities. The foil in general is either a deep clay, morafs, (out of which they dig turfs) rocky, covered with flints, and in fome places intermixed with lime-ftone, and fand. There are not at prefent, any iron works in its neighbourhood, but the remains are ftill vifible of those that were formerly wrought.

The fprings that go under the general name of Spa, are fix in number.

The Pouhon is feated at the bottom of the market-place, or - fquare, almost in the heart of the town of Spa. This appellation is derived from the old Walloon word, poubir, which

fignifies

fignifies to draw. It is chiefly from this spring that the waters are taken, which are fent into foreign countries, and befides the inhabitants use it for their common drink. The Geronftere is in a forest to the south of Spa, at the distance of about three quarters of a league. It flows out of the chink of a rock, and is received into a little bafon, covered by a dome of free-ftone, fupported by four marble columns. The Sauvoniere is half a league from Spa, to the weft: at á fmall diftance towards the fouth, is the fountain of Groifbeck, which derives its name from the Baron de Groifbeck, who in 1651 caufed it to be inclofed. The Tonnelet and Watroz, are at about half a league distance from the town of Spa, and lie to the left of the Sauveniere. These two are the leaft efteemed of the fix.

The Pouhon, and the Sauveniere, are the most antient, having been known from time immemorial; and the Author believes that one of these must be meant by Pliny, when he mentions the fountain of Tongres. The Geronftere and the Tonnelet were discovered about the beginning of the last century. Our Author acquaints us with his fentiments, as to the origin of fprings in general, in order to establish his opinion with refpect to thefe; which is, that they are not produced or affected by rains, but arife from fubterraneous vapours, condensed in the cavities of the mountains. He next, by a chemical analyfis, determines the principles of these mineral waters, and finds in them, iron, an acid fpirit, a fulphureous fpirit, an alkaline falt, a matter resembling the felenites, and air; he gives the experiments by which each of these principles was afcertained.

He then enters into fome farther detail, in respect to the decompounding of thefe mineral waters, the order in which these appearances are difclofed, and from thence explains how, from thefe appearances, may be deduced the nature and the properties of these waters. He adds to his former experiments, that the Spa-waters do not coagulate milk, that they do not make a true effervescence with acids, except with the oil of vitriol, which will act as ftrongly upon common water; that in like manner, they make none with alkalies; that they precipitate the folution of the falt of lead, in the form of milky-powder: and that fpirit of wine, and oil of tarter per deliquium, cause a weak mealy precipitation.

The temperature of these waters, it seems, is not always the fame. The Author obferves, that they varied in the fummer of the year 1756, from the forty-fixth to the fifty-second degree of Fahrenheit's thermometer. In general, the water is colder towards the bottom of the basons than at the furface; the warmer the weather, the more this difproportion appears; and the

longer

longer the water continues in the bafon, without admitting fresh, the nearer it approaches to the temper of the air.

The refult of his reafoning and experiments is, that the Spawaters are both acid and alkaline; or rather, they contain in them an acid and an alkaline; but this acid is fo fubtle, and fo volatile, that it is almoft impoffible to feparate it from the other principles. They are alfo vitriolic, that is, if we understand by vitriol, an iron diffolved by an acid; but this is fo flightly united, that the least heat, the action of the air, or the smallest agitation, produces a feparation.

These waters differ very confiderably from each other, in refpect to tafte. The Pouhon is acid and ferruginous; the Geronftere, is fulphureous, leaving an impreffion of acid and iron upon the palate; the Sauveniere has a fulphureous tartnefs, with a little tincture of the iron; the Groifbeck taftes very fharp,. fulphureous, and ferruginous; the Tonnelet is tart, a little auftere, with an aluminous vitriol, and fmells fulphureous; the Watroz has an acid tafte, and a ferruginous aufterity.

He fhews next what quantities of folid matter were left upon evaporation of two pounds of the water from each of thefe fprings. He remarks, that there is a different proportion in them of fulphureous fpirit, in which the Geronftere is ftrongest of all; after that the Sauveniere; the Groifbeck and the Tonnélet; and the Watroz and Pouhon leaft of all. It has been hitherto thought impoffible to fix, or to collect the fulphur of the Geronftere; but this Writer affures us, that he has found in the bafon a whitish kind of matter, which, when dried and fprinkled upon a hot iron, or expofed to the flame of a candle, emitted a ftrong fulphureous fcent. Thefe waters differ alfo more or lefs, in their facility of being decompofed. The Geronfterre being the moft fulphureous, is the fooneft altered, and is therefore the leaft fit to be tranfported; the water of the Pouhon, on the contrary, affords the greatest resistance, and for this reason, is that which is commonly fent into foreign countries.

The activity of thefe waters is not all, in proportion to the quantity of fixed matter which they contain; for the Geronftere, which does not hold above a third of what may be extracted from the Pouhon, is the brifkeft of all, and the most apt to occafion a giddinefs in the head; which proves that the principal virtue of these waters confifts in a volatile principle, that efcapes any chemical analyfis, and diffipates itfelf immediately into the air.

The Author, to confirm his notions, as to the nature of the waters of the Spa, and to render the manner in which they are formed

2