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ON THE CONSTRUCTION AND USE OF

CALCUTHE VALUES OF BENEFITS

to them to drip into the box; this cross of the superiority of iron over copper as pin is also useful in carrying the whole a negative plate, may refer with advanapparatus from place to place when the tage to the notices of Mr. Roberts's disbox is full of the exciting solution. covery, which appeared in the Nos.

A, fig. 4, is the box for containing the quoted by him of the Philosophical Maacid.

gazine ; as also to the Memoir on this B B, the frame plates supported by discovery read by Professor Poggendorff the cross pin going from standard to before the Berlin Academy, and Mr. standard.

Roberts's answer to the Professor's MeIn the description I have heretofore moir which will be found in the same given of the process of firing by galvanism, journal. Ed. M. M.] I have recommended that the fine steel wire of the cartridge be soldered to the horns of the cartridge communicating wires; but I now find it more convenient and equally efficient to bend the extreme COMMUTATION TABLES, FOR points of the horns upon themselves as LATING hooks; the fine steel wire is slipped across DEPENDING ON LIFE CONTINGENCIES. and under the two hooks, when a blow

Part I.-Introductory. with a hammer on the hooks closes

It is to a Mr. George Barrett, of whom them, and fixes the fine wire securely,

nothing besides is publicly known, that as thus. I need hardly mention the ne

we are indebted for the principle of the Fig. 5.

Commutation Tables, and for the method of computing, by means of them, the values of benefits depending on the contingencies of human life. The method was first introduced to public notice, after it had been refused a place in the Transactions of the Royal Society, by Mr.

Baily, in an Appendix to the second edicessity of having all the metallic surfaces

tion of his Doctrine of Life Annuities, perfectly clean. The use of the kind of published in 1813. Mr. Griffith Davies, conducting wire explained in sec. 72,

in a work on life contingencies, published greatly facilitates the operations of blasting.

in 1825, by certain additions to the Tables, If these operations are on a large scale,

and alterations in their structure, accordit is better to cover the conducting wire ing to Professor De Morgan, “ increased with twine than with cotton, as it is more

the utility, and extended the power of durable, and ensures a better insulation

the method to an extent of which the inof the wires from each other.

ventor had not the least idea." Mr. BarIf " W. C.” intends fitting permanent rett's method was also briefly noticed in batteries in his works, I inay perhaps be

the Appendix to Mr. Babbage's Treatise able to point out to him some changes

on Life Assurance. The method, as imnecessary for economy in such a mode proved by Mr. Davies, has since been of working the battery; and as it is pos

treated, and a very large collection of sible I may shortly be in the neighbour

Tables adapted to it, for both one and hood of Liverpool, if “ W.C.” will send

two lives, has been given, by Mr. Jones, me his address within a few days, I will

in his work on Annuities, in the Library endeavour to call at his works, and be of Useful knowledge. But by far the happy to give him all the information I

most valuable papers on the subject are can on this subject. I trust that all your

two in the Companion to the Almanack, readers that use my process will give a

for 1840 and 1842, by Professor De detailed account of their operations in

Morgan, which contain the materials of your pages.

many thousand formulæ, applicable to I am, Sir, yours, &c.

almost every case that can occur. There

is also some notice of the method in the Martyn J. ROBERTS.

article Reversions, in the Penny CyOctober 17, 1842.

clopedia, which article is likewise the [Such of our readers as may be de- production, we believe, of Professor De sirous of knowing more about the cause Morgan.

The above-named are understood to that the present papers are intended; be the only works in which the new me and if they shall serve to render more thod is, in any sort, treated. They cer available than heretofore, to any of the tainly are not numerous; but they are numerous readers of the Mech. Mag., the sufficiently so, and well enough known, valuable papers to which reference has so to have induced a general adoption of often been made, the writer will consider this method, to the exclusion of that pre himself amply rewarded for his labour. viously in use, but for two reasons. The While he thinks that the task of renderfirst is, the want of Tables adapted to this ing more generally intelligible the prinmethod ; and the second, the want of an ciples on which the new method is elementary and systematic treatise on the grounded will not be a difficult one, he subject. The first of these wants is now trusts that, should his success not be amply supplied by the valuable collection commensurate with his wishes, he may of Tables published by Mr. Jones, above at least be found to have aided in clearreferred to; but the second still exists. ing the way for some one better qualified It is no disparagement to the able authors to do justice to a subject, which is daily of the works above named, to say that growing in interest and importance. this is the case.

Generally speaking, it The peculiarities of the old and the has not been their object to furnish a new methods may be here briefly stated. treatise of this kind; and they have ac In the old method, we are presented with cordingly taken for granted, on the part a table of the values of annuities at all of their readers, the possession of a de ages, which of themselves are rarely gree of acquaintance with the subject, wanted, but from which, by operations which very many, to whom the power of more or less complex, the values of beusing the Commutation Tables would be nefits of all other kinds may be comof the greatest service, certainly do not puted. In the new method, on the other possess. Professor De Morgan, indeed, hand, we are presented with a table expressly says, that all that will be found which, by mere inspection, tells us noof demonstration in his articles " is in thing; but from which, while the values tended for those who are familiar with of the ordinary benefits can be found by the subject." Now, although the for a simple division, those of benefits of the mulæ, according to this method, are ex most complex description are found by tremely simple, and easily intelligible to operations consisting usually of nothing any one who is acquainted with the merest more than one or two subtractions and rudiments of algebra, and who will take one division. In point of simplicity, the small degree of trouble necessary to moreover, in the deduction of the various enable him to comprehend the notation formulæ, the methods admit of no comemployed; yet it is a result of our own parison. For the establishment of what, experience, which we have no doubt can according to the old method, required be amply confirmed by the observations of chapters, a few pages will suffice accordothers, that most people view with mis- ing to the new. trust, and will not willingly have recourse The opinion we have expressed as to to formulæ, the principles of which they the superiority of the new method, how. do not understand. And while Professor ever, will probably be regarded as of De Morgan's papers, in particular, have, little value; and that of Professor De doubtless, well served the end which, Morgan, which is most unequivocally from the remark quoted above, the dis given, may perhaps be objected to, as tinguished author appears to have had the testimony of an interested witness, principally in view, we do not see that seeing he has bestowed such pains on the this forms any good reason why others, elucidation of this method." But Mr. to whom a knowledge of this method of Milne's testimony will certainly not be computation would be of service, should objected to on any such ground. He is be prevented from availing themselves of the author of the best treatise that has the vast fund of information, regarding appeared, or is now likely to appear, on this thod, which these papers con

the old method ; and therefore his pretain.

judices, if he has any, must be supposed It is, therefore, as an humble contri to be all on the side of that method which bution towards the supplying of the de he has done so much to illustrate. Speakficiency, which has been shown to exist, ing of Mr. Baily's work, he says, (En

to say,

cyclop. Brit., seventh edition, article, age will survive a year. A preliminary ANNUITIES, Vol. 3, p. 200.) “ In an step therefore was, by means of these proAppendix to it *

. formulæ were babilities, to construct a mortality table of given for calculating from tables of that the more usual form ; and in doing so, as kind [Commutation Tables] the values well as in the subsequent construction of of temporary and deferred life-annuities our Commutation Table, Mr. Finlaison's and assurances, when the annuity, in data have been made use of to their full stead of remaining always the same, in extent. It is thought proper to mention creases or decreases from year to year this, because there is an abridged" by equal differences, with considerably mortality table of the usual form, deduced greater facility and expedition than the from Mr. Finlaison's data, published in same things could have been done with the Report of the Select Committee on by the tables and methods of calculation Friendly Societies, 1827, page 82; and in previous use." And his testimony, some might be embarrassed by finding be it observed, refers to the tables as that the results of our Commutation Tadevised by Mr. Barrett, and gives but a ble do not exactly correspond with those faint idea of their capabilities in their im which may be deduced from the abridged proved form.

table referred to. This table states the We might now proceed to describe the survivors of 1,000 births to be, at the Commutation Tables, referring for illus ages of 6 and 7, for instance, 919, and tration to one which the Editor has kindly 912, respectively; while the use we have agreed to insert. But as it will be more made of the data enables us convenient that the table should accom that the survivors at these ages, out of pany the next paper, we prefer here dis 100,000,000 of births will be 91,912,811, posing of some preliminary matters. and 91,239,410. As regards the correct

The construction of the Commutation ness of our Commutation Table, it has Tables is effected by combining in a par. been subjected to the severest tests in the ticular manner (which will be explained way of verification, and we are confident in the next paper,) the rates of mortality that no error of any consequence will and interest; and, as in the tables adapt be found in it. Mr. Finlaison, in his ed to the old method, any rates that are Report, gives the values of annuities at most approved of, as regards these ele all ages, at 4 per cent., as deduced from ments, may be employed. But whatever his data by the ordinary method, and carmay be the rates of interest and mortality ried to 7 places, which are two more than made use of, the demonstrations and for are usually given. The values derived mulæ which will be hereafter given, being from our table will be found to corresgeneralized by the employment of sym- pond with those to the last place. It has bols, will be equally applicable to all been thought proper to make these retables of the same form.

marks as to the degree of confidence that The rate of interest according to which may be reposed in our table, in case any our table, (which has not been heretofore one should feel disposed to apply it to published,) is constructed, is 4 per cent.; practical purposes. and the rate of mortality is that given by We shall now explain the notation, as Mr. Finlaison in his Twentieth Obser regards the rates of interest and mortality, vation on the Mortality of the Govern which we shall employ in our demonment Annuitants (Parliamentary Paper, strations. No. 122, 1829, page 58.) Tables of It is shown by writers on interest, and mortality in their usual form exhibit the indeed in most elementary works on Alnumbers who, out of a number supposed gebra, that if r represent the interest of to be alive at birth, or some other early £1 for a year, then will age, attain each successive year of age, 1+p be the amount of £1 in 1 year. and consequently also the nuinbers who (1 + r)?

£1 die in each year, and this form is the (1 + r)s

3 most convenient for the construction of (1+r)

LI Commutation Tables. But this is not and generally (1+r)* will be the amount the form in which Mr. Finlaison's data of £1 in x years, where æ may represent are arranged. What he gives are the any number whatever, all at compound probabilities, or rather the logarithms interest. It is also shown, that unity diof the probabilities, that a life at each vided by the amount of one pound in a

2 years.

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2 years.

given time, is equal to the present value would, in the given time, just amount, of one pound, to be received at the end also at compound interest, to one pound. of that time; that is, to the sum which Hence

1

is the present value of £i due at the end of 1 year.
1tr
1

£1
(1 + r)2
1

£L..

3
(1 + r)
1

senting the number alive at birth, which and generally

is the present (1 + r)

is called the radix of the table, will, in

the case of the Carlisle Table, be equal value of £i due at the end of x years, X,

to 10,000, and in that of the Northampas before, denoting any number whatever.

ton Table, to 11,650 ; and the values of

1 It is usual to denote by v.

the other symbols will in like manner 1+r

vary. But the great advantage of the

use of symbols is, that we have no need Hence, the foregoing present values will

to distract ourselves with their particular be more conveniently represented by

values, until we reach the final solution V, va, v), .... v*, respectively. It is ob

of the problem with which we may be vious that the present value of any other

engaged. Also, since the number who sum, will be found by multiplying the

die in any year of age, that is, who enter present value of £1, due at the end of

upon that year and do not live to comthe same number of years, by the num

plete it, is equal to the difference between ber of pounds in that sum. Copious

ihe number who complete that year and tables of the present values of 11., due at the number who completed the previous the end of any number of years, from

year, these decrements, as they are called, one to one hundred, together with their

will be represented as follows :logarithms, are given in Mr. Jones's

1o-1, is the number who die in their 1st year, work, Part I.

li-la

2nd The indications of the mortality table 12-13

3rd are represented as follows. The number

lx-, -bx

th shown by the table to attain to any age is

(r+1)th denoted by the letter l, with the age attached as a suffix. Thus, lo, lv, la, la,

and so on, x representing any age what

ever. Ix, &c., denote the numbers who attain the successive ages, 0, 1, 2, 3, X, &c.,

We are now in a condition to enter respectively. These symbols will, of

upon a description of the Commutation

Tables, and their construction; but this course, denote different numbers in the case of different tables. Thus lo, repre

we shall reserve for our next paper.

G. Hermes-street, Pentonville.

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COMBUSTION-THE CRANK DISCUSSION,

Sir,--As I was shutting my windows a few days ago, to keep out the visible and tangible particles of a metropolitan fog, the blackening influence of which, even the town sparrows bear witness to, when compared with their congeners in the country, I could not help thinking what a benefactor indeed, (I do not mean to the sparrous), Mr. Williams would be, if he could improve the combustion in our common grates, in the same degree as he has improved furnaces on a large scale.

It must be evident to every one, that has

ever poked a fire, that large quantities of unconsumed gas, as well as smoke, are con. stantly escaping, only to add to the murky atmosphere without, so common here in the cold seasons of the year ; gas in fact constitutes the greater part of what we commonly call smoke : and I believe it is also incontrovertible, that none of the most improved stoves hitherto invented, have succeeded either in burning all this gas, or in preventing the generation of smoke, during the combustion of that part of the gas which they do burn : yet surely when this object

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has been attained in a furnace, we might expect to find it also attainable in an open grate. Mr. Williams's exact principle could not perhaps be brought into action, but might not an equally efficient one? Even burning all the gas would be a great improvement.

The origin of the form of our present grates has been evidently more a desire to fit them for throwing out heat, than any consideration with regard to attaining a perfect combustion : till of late years, all that was thought to be necessary, was to “put your combustibles in a convenient place, set fire to them, and let them provide themselves with air.” Now on looking at a common fire-place, (particularly when a fire has been lately replenished), we perceive at once that almost all the smoke comes from the back and sides of the grate, and that the gas will not burn well in either of those situations ; burning, when it does ignite, generally with a lambent Jack o’lantern flame, ready to die away every instant, from want of air ; whereas the gas evolved in the fore-part of the grate burns brilliantly and without intermission. This seems to suggest that the grate should be all front, (being an Irishman, Sir, you know I am privileged—so no criticizing, if you please), or at least that such a state of things is a desideratum to be approximated to, as nearly as may be, without disregarding other requisites. This perhaps is already the case in several newly invented grates, as far as the sides are concerned, but I do not think I have ever seen a grate in which the back admitted a good supply of air.

I have said nothing, Sir, as to any means of preventing bona fide smoke, and for a good reason, because I am not aware of any; experiments in these matters require such “appliances and means to boot,” that it is not everybody that can try them ; but it would be well worth the while of any of your readers who have it in their power to do so. Perhaps this may meet the eye of Mr. Williams, and induce him to turn his attention, if he has not done so already, to this branch of a subject upon which he has already thrown so much light; even if it were only in pity to us poor metropolitans, condemned for half the winter to have the sun no brighter than a new copper penny.

I am, Sir,
Your obedient servant,

R. W. T. P.S. I may take this occasion, Sir, to remark that your note appended to your correspondent “ M.'s" last letter on the crank, would have been quite sufficient to stop any reply on my part, had I intended making one; but I beg leave to assure you,

THE TREASURE SHIP LE TELEMAQUE."

We learn from the Journal du Havre, that the works for the salvage of Le Télémaque are steadily advancing. The little town of Quilleboeuf is quite alive with shareholders and speculators, who daily arrive, to satisfy themselves as to the progress of the undertaking, and to watch the market for shares, which mount in price as the long-buried wreck gradually rises from the sand.

The works having sufficiently advanced, the process of weighing was commenced on the 14th of October ; and after every thing was tightened up, a stress of above 600 tons was applied, by means of powerful screws acting between the bridge raised on the vessel and the large platform constructed on the rocky bed of the river. The experiment was anxiously watched by all present, and no small uneasiness was felt by some, who saw the enormous timbers bend and creak, while the wreck appeared still as firmly fixed in the sand as ever. After about twenty mi. nutes' screwing, the operations were suspended, and the vessel left to the operation of the current under this great strain. In about six hours the timbers had straightened; and when the screws were then again applied, the wreck, with the vast superincumbent mass of iron and timber, yielded to the force, and, once free from the suction of the sand, the whole, though of very great weight, became perfectly manageable.

The current having thus done its work in loosening and clearing the wreck, it became necessary to protect the works from its force, and additional piles were accordingly driven, to form a small breakwater, in case of storms. The weighing has also proceeded as fast as due precaution would allow; and by the close of last week, a workman had walked along the wreck, having the water only to his waist.

It will still demand some days to raise the Télémaque entirely above low water mark, to support her from beneath, and then to cut through all the chains and timbers to open and discharge her. The interest felt in France, now that the engineering question has been solved, is unprecedented. The de. mand for the shares is very great; but though a few may be picked up at from 101. to 201. each, the majority of the holders refuse to sell under one thousand francs for the hun. dred franc share.

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