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To the Editor of the Monthly Magazine.

SIR,

T appears to me that our present scarcity of gold is in a considerable degree attributable to its bearing in this country, when coined, a nominal value considerably less than its real value; a piece of gold of the size and weight of a guinea, being worth at least twenty-four shillings, while the guinea itself passes for only twenty-one shillings. This far nishes an inducement to Jews, smug glers, and others, who usually engage in contraband traffic, to export the coin of the kingdom, while on the other hand there exists no encouragement to import the metals of which it is made; for the exporter of a guinea receives for it abroad at least twenty-four shillings, though he has taken it here for only twenty-one shillings; but the importer would have to give twenty-four shillings for a piece of gold of the same size, for which the government here could only afford to give him twenty-one shillings, since the piece of money they would make of it would pass for no more than that sum: and that twenty-four shillings is the real value of such a piece of gold, is evident; because that is the price given for guineas, when collected for the purpose of exportation, and because they must fetch even a higher price abroad, in order to yield a profit proportionate to the trouble, expence, and risk, of collecting and exporting them.

It is obvious, therefore, that as purchasers of gold, we offer a less price for it than other nations; that their markets are the best for the sale of it; and that, while they continue so, all the gold will be carried to foreign markets in preference to ours.

The same observations are applicable to our silver coin, the scarcity of which has arisen from the same causes as the scarcity of gold, with the additional cause arising from the necessity our merchants are under, of paying for teas, and other Chinese commodities, in that metal only. Hence it is evident that our coin is disproportionately large, and that, while it continues so, the advantage of purcha. sing gold and silver here at a certain rate, and of selling them abroad at a much higher price, must have the effect of enCouraging the exportation of them, and consequently of encreasing their present scarcity, while the evident loss on the importation of them to so bad a market, will stand as an insurmountable obstacle to the replenishment of our mint by the MONTHLY MAG. No. 213,

only means by which it can be replenished, namely importation.

These reflections lead directly to the means by which the further decrease in quantity of our gold and silver coin may be prevented, and their return to this country facilitated. Instead of offering only at the rate of twenty-one shillings per guinea for gold, and twelve pence per shilling for silver, let our government offer the same price for them as they fetch in foreign markets, or even a little more, and issue a coinage of a corresponding weight and size. This will at once destroy the inducement to export our coin, and encourage the importation of the precious metals into this country.

That other causes have tended to create the present scarcity of gold and silver, I am fully aware; causes whose effects have been infinitely more extensive, and infinitely more ruinous. [ know that the enormous subsidies with which we have, at various times during the last twenty years of war, paid foreign princes for fighting their own battles, and the immense sums we must have remitted abroad for the payment of our armies, whilst engaged on foreign service, must have drained us of millions and millions of specie: but these are causes the effects of which I fear are irremediable.

I know that twenty-four shillings is the price given for guineas for the purpose of exportation, for I was accosted about a month ago, at Dartford, by a Jew, who was returning from a journey made expressly to collect them, who asked me whether I had any guineas, and said, if I had, he would buy them of me at that rate. He told me he had purchased 165 within the last three days, and that he had given a one pound note and four shillings a piece for them. I asked him how he could afford to give so high a price, and he answered in a whisper,

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They are for exchportation, and you may be sure I makes it vorth my vhile." At the time, I thought it was only a Jewish expedient to pass a forged note, or to exchange four bad shillings for one good one; but I have since learned that the price be offered for guineas was what they were really worth.

April 13th, 1811.

H.

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be printed from a copy corrected by the author, and carefully revised by L. Gwynne, A.M. master of the Royal Mathematical School, Christ Church, London, &c. &c. and, observing that a very material error still remains uncorrected, I beg, through the medium of your very useful publication, to point it out, for the benefit of such persons as may use the tables in the above work. 'The error I allude to is in the applica. tion of the second part of the equation to equal altitudes, (in Table II.) as explained by an example in page 81, which the author prefaces thus, "Another example will make every thing relative to these tables perfectly plain to the meanest capacity.'

In this example the latitude is 83° 56' south, and the first part of the equation 10 64 which is subtractive, because, the latitude being south, the sign is changed from+to-. But we are told that the second part of the equation 17:25, is subtractive likewise, naturally leading one, not of the meanest capacity, to suppose that the signs in Table II. are to be changed when the latitude is south, which is contrary to the construction of the table, and will certainly produce an erroneous conclusion in the calculation; for in the present instance the whole equation is said to be 11"-89, instead of which it ought to be-9"-39, being the difference instead of the sum of the two parts of the equation, and subtractive because the greater part is so.

M.

To the Editor of the Monthly Magazine.

I

SIR,

8

HAVE read in your last Number communication entitled "Objections to Doctrines adopted by Mr. Baily and other Writers, on the Probabilities of Life," by Nathaniel Hawes, on which I beg leave to send you a few comments,

Mr. Hawes first briefly mentions the manner in which the probability of life has been expressed by Halley, De Moivre, Simpson, Dodson, Price, Mor. gan, and Baily; and then states, that the purport of his letter is "to represent the fallacy of such a doctrine." This certainly is a very modest beginning, and I assure you that, when I had proceeded only thus far, I formed no very elevated opinion of the ability of your corre spondent; for, though I by no means think it proper that we should place an implicit confidence in names, particularly when we may, by the exercise of our

own reason, ascertain the truth or falsehood of whatever has been delivered concerning the subject of our enquiry; yet when men, pre-eminently distin guished for their abilities, and for their arduous enquiries after truth, have suc cessively maintained a doctrine by the force of reason and argument alone; I think it must be confessed, that suddenly to renounce such doctrine, and to pronounce it fallacious, argues a degree of self-confidence which is nerally found to accompany vanity and ignorance, than truth and knowledge.

more ge

Mr. Hawes next gives a very obscure and inelegant definition of what he "takes to be” a fraction; he then proceeds thus: "By consulting Nature in preference to my own imagination, or to any received doctrine, I find the probability that a person, whose age is twenty, shall attain to the age of fifty, or live thirty years, is, according to the obser vations of M. De Parcieux, as given in Mr. Baily's third table, equal to 25.6689

30.0000

years." But why, Mr. Editor, did not your correspondent acquaint his readers with the method by which he found the probability that a person, aged twenty, should attain to fifty, was 25.6689 S0-0000

Not one word however has he

said of the modus operandi. No, but he says that he has consulted Nature: true, I know he says that he has consulted Nature; but may not his reader be at liberty to doubt the truth of this assertion? For does it not seem strange that Nature should have thrown her whole

blaze of light upon Mr. Hawes, and have afforded only a few occasional rays to Newton, Halley, and De Moivre; rays too, which it should now seem, only served to bewilder and deceive them? Is it not extraordinary that Nature should have been so munificent of her favours in, most probably, her first interview with your correspondent, and that she should have been so coy in her manners, so reserved in her appearance, and so niggardly of her gifts, to those great men, who spent their whole lives in her so ciety? Surely, therefore, Mr. Editor, your readers may be at liberty to doubt this consultation with Nature, and to rank it in the class of those experiences, as they are called, which are not un frequent among the members of a certain religious persuasion, but which are some

times

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finds, from the same consultation, to express the probability that one only of those persons would be living at the end of the said time; that is to say, that it is more likely that two persons should be both found alive at the end of any given time, than that one of them only should survive to the end of the said time! Mr.

Hawes further says, that he finds (whe

ther from the same consultation of Nature, or from any after interview with the goddess, he has omitted to state) that it is more probable that three persons, whose ages are 20, 30, and 40, should all be found alive at the end of fifteen years, than that the person whose age is 40, should alone be found alive at the end of that time; for he finds the fraction 13.0505

15 0000

to express the probability that

12.5836 15 0000

the

all of them will continue so long, but only the fraction to express probability that the person, aged 40, will continue to the same period. And thus, Mr. Editor, from that glorious in flux of light which has fallen upon us, through the liberality of your correspondent, we are now to believe, that, if there be a hundred persons of the same age, it will be more probable that they should all be found alive at the end of any specified time, than that some one of them only should be found alive at the

end of the said time. One almost wonders how such an instance of absurdity could have escaped even Mr. Hawes himself, accompanied as it is with sneers of contempt upon the most accurate of all sciences, the mathematics, and upon the ever-to-be-revered names of Halley, De Moivre, Simpson, Dodson, and Price.

But lest our minds should not be suf ficiently illuminated by the light which Mr. Hawes has shed upon us in the communication of the results of his consultation with Nature, which have been above stated, or rather, perhaps, that his readers might entertain a due sense of the high estimation in which Nature held this, her darling son, from the length of her conference with him; this gentleman has added, as a conclusive argument for the ignorance of Halley, De Moivre, Price, &c. &c. that he finds (from his consultation with Nature, mind ye) that the probabilities of a person aged 15, continuing in being 10, 20, 30, 40, 50, 60, 65, 70, and 80, years, will be respectively *9 5837, 18-2394, 25-9894, 32-8101, 38-2624, 41.8909, 42.8573, 43.3278, and 43.5094; thus making the probability that a person should be found alive at the end of 80 years, nearly five times as great as the probability that the end of only 10 years. Bravissimo, Mr. same person will be found alive at the Hawes! Bravissimo Daine Nature, Mr. Hawes' confidential adviser!

Your correspondent concludes his modesty with which he commenced it; communication with the like expressive ceeded" in representing the fallacy of a for he says, that he trusts he has suc doctrine so confidentially authorised, so mathematically tolerated, and so impli citly acquiesced in, during the last hundred years;" and, by way of climax, closes with, "It is only left me now to enquire, on which side of the question, conviction preponderates?"-Yes, Mr. Hawes, you need not for one moment doubt but your readers are fully con vinced, that the "suggestions" of Dr. Halley were foolish; that the "adoption" of those suggestions by Mr. De Moivre affords a proof of ignorance, the "adherence" to them by Mr. Simpson, an astonishing instance of obstinacy; the "confidence" placed in them by Mr, Dodson, a display of rashness; the espousal" of thein by Dr. Price, an

66

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indication of imbecility; the "embrace" of them by Mr. Morgan, an example of the effect of dotage; the "assent" to them by Mr. Baily, a manifestation of good-natured credulity; and, finally, the "conviction of the fallacy of their doctrine, by Mr. Hawes himself, a splendid and memorable instance of the effica cious power of nature when properly consulted.

I cannot conclude, Mr, Editor, with out observing that, in the " objections" of Mr. Hawes, the name of Mr. Baily holds a conspicuous place. Why this gentleman's name should have been so frequently mentioned, I cannot conceive, unless with a view to depreciate the value of his works; permit me there fore, Sir, to state, that I have read and studied the greater part of Mr. Baily's work, on the "Doctrine of Life Annuities and Assurances," and that I have no hesitation in declaring it to be by far the most excellent performance which we have on this subject. In the theoretical part of this work, by his great skill in analysis, and by a more happy notation, the author has demonstrated the principles of the doctrine of annuities, in a manner which delights, no less by its elegance, than by its scientific accuracy; and in the practical part, the very extensive and appropriate list of examples, and the valuable collection of tables, while they display the unwearied exertion of the author for the perfection of his work, render his book of the highest utility, not only to all the Assurance Companies in the kingdom, but also to every individual who has any interest in, annuities of every kind, or in the renewal of leases. Norwich, April 17, 1811.

R. SAINT.

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Hawes's system, which, to any person conversant with the science, must, at one view, appear unfounded, if I did not think it essential that there should not be two opinions on so important a subject; as, you must be aware of the great extent of business that is daily transacted in this metropolis upon the principles that it is Mr. Hawes's object

to overturn.

Mr. Hawes has very sedulously kept us in the dark, as to the foundation upon which his superstructure is raised; and the only effectual mode of showing its folly will be, by contrasting it with the simplicity and clearness of the doctrine laid down by Dr. Halley, and the other authors named. He asserts, (to use his own words,) that the subject of the present investigation is that of time, that is, its component and fractional parts:" now it appears to me evident, that not "time," which is made up of divisions fixed, and not subject to mutation, but the probability of a given event happening, or not happening, in any one, or more, of those divisions of which time is composed, is the point in question; and that given event being death, we can only determine the pro bability of its happening, or not, by a reference to those tables, that show the progression in which given numbers have died off, from birth, to the latest probable period of human existence.

The fraction that gives the probability of a person being in existence at the end of any term, as expressed by every author who has treated on the subject, is this, the denominator shews the number taken from a table of mortality, living at the age of the person and the numerator, the number living at an age older than the given age by the term stated: the reason of this may he given in few words; in any table of mortality the number, therein stated to be alive at a given age, shows the num ber of chances for person of that

age, both living and dying, in any term; the number alive at an age older, by that term, than the given age, shews the number of chances for living to the end of the term; and the difference between those two numbers, shows the number of persons that die in the term, or the chances for not living so long. This will be made perfectly familiar by an example; let it be required to find the probability that a person, aged 20, shall live 30 years, (as in the first example quoted by Mr. Hawes) and also the probability that he shall

not

not live 30 years; the number living at the age of 20, in Mr. Baily's third table is 814, and at the age of fifty 581the fraction, therefore, expresses the probability that the person in question will be alive at the end of that term, and the fraction 1 shews the probability of dying in that time: both those fractions added together or 4X

4 will be equal to unity, as it is cerLain that the party will be either alive or dead at the end of the term.

I should have been at a loss to discover in what way, Mr. Hawes obtains his numerators, if I had not observed that in those examples, where he makes the term for which the probability is to be found, equal to the difference between the age given and the oldest age in the table of mortality, from which the calculation is to be made, the numerator is the same number that is given in the table of expectations deduced from the same table of mortality; as, for example, he makes the probability that a person, aged fifteen, shall live eighty years, 4:5833 the numerator of which fraction is the number given in Mr. Baily's third Table, as the expectation of a life of fifteen; and from hence I conclude, that the mode which he so confidently offers as a substitution for the present, is nothing more, than in the case of single lives, the making the number of years, for which the probability is to be calculated, the denominator, and the expectation of life for that term the numerator. By the expectation of life, I mean the share of life, which, according to any table of observation, belongs to any individual of a given age, or in other words the average number of years which they will, one with another, enjoy.

Mr. Hawes has forgotten, that by attempting to overturn the present mode of estimating the probabilities of life, he tries to overturn the way by which his own numerators are calculated, which are nothing more than the sums of the fractions expressing the chances of living one, two, three, &c. years, to the end of the term named. In the case of joint lives, I need not under. take any investigation, but shall content myself with observing that he has been guilty of a palpable error, which at once shews his whole system to be founded in absurdity, and maintained by ignorance. I mean, his making the probability that two lives shall continue together in existence to the end of a term, greater than the probability that

one of them shall live to the end of
In his first example
the same term.
of his own method, he makes the chance
of a person, aged twenty, being alive at
the end of thirty years 35:6683 years:
that a person, aged forty, shall be alive
at the end of thirty years, 33:3855 years,
and the probability that both shall con-
tinue in being together to the end of
the same term 33:6588!!!~The same
result is produced in every example he
has given.

correct.

It is now almost time for me to leave Mr. Hawes, whose futile attempts will avail little, in opposition to the doc trine laid down by such men as Hailey, De Moivre, Simpson, and other eminent authors, and so ably treated by Mr. Hawes's contemporary Mr. Baily, who has certainly combined in his va luable treatise on the subject nearly all the information to be found in preceding authors, in addition to his own improvements, althought it is to be wished that he had blended with his talent some greater portion of liberality. I would however, before I close, inquire of Mr. Hawes in what way writers, on this branch of science, have overstepped the bounds of probability, and why their principles are not Mr. Hawes seems by his sneers, at what he calls the "mathematical faithful," to suppose that the science may be made independent of the mathema tics; to that I shall observe, that haď he been able to investigate the subject mathematically, five of your columns would not have been occupied with the tissue of absurdities, we have seen from his pen; nor would he have asked so many unmeaning questions, which have no other tendency than tổ perplex his readers and to involve the By what ima question in obscurity. ginary law of Nature does Mr. Hawes make his deductions from registers of Can he suppose that life and death? a system laid down by the authors he has mentioned, will yield to his insig nificant attack, which is unsupported by either reason or argument? your correspondent stated his objec tions, with becoming modesty, and deference to acknowledged talent, and manner free offered his system in a from arrogance, he might have been considered ingenious, or at the worst have passed unnoticed; but his style is such as cannot fail of exciting emotions of contempt for his vanity, and pity for his ignorance.

Had

PHILO MATHEMATICUS,

To

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