It may be here remarked, that the Northampton Tables of the duration of life are not at all adapted to Iver M'Iver's question, and if practically acted upon would prove ruinous to the proprietary, whether the interest be 3 or 4 per cent. For it is now universally admitted, that the Northampton Tables of the duration of life make the waste of life much greater than facts warrant. Such tables, however, have a tendency to produce great profits to the proprietary of an Assurance Office in the ordinary cases for insuring lives for any given sum and period, or for the whole duration of life. But those

Tables that produce great profit to the proprietary in this mode of assurance, will be found to produce an exactly contrary effect in the application of the scheme proposed by Iver M'Iver; or in other words, if the annual premiums were computed from more healthy tables than the Northampton, then the said premiums would be found to be much higher than those deduced from the Northampton Tables. But this, Mr. Editor, as well as some other calculations required by your correspondent, I will make the subject of another communication.

I am, Mr. Editor, yours, &c.

GEORGE SCOTT.

Cochrane Terrace, St. John's Wood.

THE CASE IN LIFE ASSURANCE-ANOTHER

SOLUTION.

Sir,-Whatever may be the qualifications and acquirements, in other respects, of your old correspondent Iver M'Iver, it is pretty clear that he is but a tyro in the matter of Life Assurance. Were he otherwise, he would hardly have called the attention of your readers to a question so very simple as that contained in page 95 of your present volume, or trammelled it, as he has done, with conditions which have nothing whatever to do with its solution.

Thus, he says-" One hundred individuals, each of the ages of twenty-three, subscribe annually for ten years a sufficient sum each to enable the survivors to receive each 1007. at the end of twenty years." And then he asks, how much should this annual payment be, according to "the Northampton or any other tables of probabilities ?" Now on this I remark, first, that it is of no consequence

whether the number of assured be a hundred or a hundred thousand, since the solution of all such questions proceeds on the supposition that the mortality to be experienced will be precisely in proportion to that indicated in the table which is assumed as the basis of the calculation. Seventy thousand survivors out of a hundred thousand will give the same result as seventy out of a hundred. Secondly, he wishes the result according to the Northampton "or any other table." Does he not know that each table will give a different result? Or does he wish the result according to all the known tables, such as the Carlisle, the Chester, (male and female) the Government (male and female), the Equitable, the Amicable, the Swedish, &c. &c.? Could he not have contented himself with specifying one table, and then an answer according to his wishes could have been given.

Lastly, he desires to know what the annual premium would be, "supposing the proprietary to receive one-third of the profits?" Here your correspondent betrays his ignorance most of all. Does he not know that the very principle on which the solution of all cases of life assurance and annuities proceeds, is, that there will be no profit on either side; that the present and prospective payments of the assured will be an exact compensation for the prospective payments of the assurers? If Iver M'Iver will first tell us what addition is to be made to the premium in name of profit or commission (a practice almost universally followed by the offices to pay expenses, and guard against fluctuations), or inform us of the deviation which is to take place in the mortality of his hundred individuals, from that indicated in the table which we are to take for our guidance, then, but not before, we shall be able to gratify him. The question, divested of what does not belong to it, is as follows:Required the annual premium, payable ten times in all, the first payment to be made immediately, subject to the failure of the life, to entitle an individual, now aged 23, to 100%. on his attaining the age of 43, interest being reckoned at three per cent., and the rate of mortality according to the Northampton Tables.

I shall solve this question without reference to the doctrine of probability. By the Northampton table 3,404 is the

number who, out of 4,910, alive at 23, attain the age of 43; 3,404., therefore, is the sum which will have to be provided to pay each of the survivors one pound twenty years hence. But since money bears interest at the rate of three per cent, the present value of one pound to be received twenty years hence is

1

10320553676.

Therefore the present value of 3,4047. to be received twenty years hence is

3,404 × 553676=1,884 713104, and this is the sum which would have to be paid now to provide one pound for each of the survivors at the age of 43. Moreover, as the 4,910 individuals all contribute equally, the division of this sum by the number of contributors gives 383852, the present contribution of each. And, finally, multiplying this by 100, we have 38 3852=381. 78. 8d. for the sum to be paid down now by an individual aged 23, to secure 1007. on his attaining the age of 43.

But this benefit is to be purchased, not by a single payment, but by an annual premium for ten years, of which the first payment is to be made immediately; that is, by a temporary annuity upon the life of the individual for nine years, together with a present payment of that annuity. First, let us find the value of an annuity of 17. paid in this way. To do this we must find the present value of an annuity of 17. deferred for the same number of years. This value is equal to the value of a life annuity at the advanced age, that is at 32, multiplied by the number alive by the table at that age, and by the present value of one pound due at the end of the period of deferment, and the product divided by the number alive at the present age. Taking the numbers indicated from the tables, we have

16.5398 x 4235 x ⚫766417

10.9337

4910 Now since the sum of the present values of an annuity deferred for nine years, and of an annuity to continue nine years, is evidently equal to the present value of a life annuity; therefore, taking the value of the life annuity at age 23 from the table, and subtracting from it the value just found, we have

18.1486 10.9337

which is the value of a temporary annuity

The best and simplest method of solving all questions relating to life contingencies, (and the shortest, withal) is by means of tables of a certain construction, of which a copious collection is contained in Mr. Jones's "Treatise on Annuities,” in the "Library of Useful Knowledge.' Mr. Jones calls the tables alluded to, Preparatory Tables; but Professor De Morgan, who is the writer of two able articles in the Companion to the Almanac, for 1840 and 1842, explanatory of their construction and applications, (which articles, I fear, are not so well known as they deserve to be,) calls them, with, as it seems to me, more propriety, Commutation Tables.

As an example of the application of the tables I have referred to, I shall, with your leave, Mr. Editor, give a solution according to them, of the problem I have solved above in the usual way. I must premise that the tables consist of various columns headed D, N, s, &c., respectively; and that the number in any column, opposite any age, is denoted by the letter at the head of the column, with the age appended to it, either as a suffix, or enclosed in brackets. I shall adopt the latter mode, for the convenience of your compositors.

Let the present age be r, the sum assured s, the number of years to elapse before the sum assured becomes payable n, and the annual premium, payable 2, times p.

Whence p

=

N (x-1)-N (x + l − 1) This is a general solution, applicable to any age, any period of deferment, any sum assured, any number of premiums, and any table of mortality and rate of interest for which commutation tables have been calculated. To apply the formula to the case before us, restore the values of x, n, s, l, and it becomes 100 D (43) N (22)-N (32) and taking the numbers from the Northampton 3 per cent. Table in Jones, (p. 242,) we have finally—

p =

MR. VALLANCE ON THE NIGER EXPEDITION, AND HIS PNEUMATIC RAILWAY.

Sir,-An error of your printer, which makes my letter to Captain Trotter, R.N. (the copy of which is given in last week's number of your Magazine) appear as if it had been written only a month instead of two years ago, renders it necessary that I should solicit the favour of your stating that it was sent in June 1840, instead of the present year.

Being compelled thus to trouble you, I take advantage of the opportunity to ask the additional favour of your inserting the subjoined paragraphs from a paper which I wrote six years ago, in

order to counteract an erroneous statement, and what I deem an erroneous opinion of Colonel Macerone's. That gentleman states that "the passengers had no other light than that of lamps" in my pneumatic railway; and that he considers "" super-terraneous" conveyance the best. The reasons assigned in the subjoined paragraphs render me of a different opinion.

I am aware that I stand alone in this belief. But as it is the result of the conviction which makes me unwilling (not unable) to bring forward any method of going above ground, such as Messrs. Medhurst, Pinkus, and Clegg have published, all of which I believe myself prepared to prove, may be superseded by better methods being made practicable. I am unwavering in the preference which, for so many years, I have given it; and am willing to risk any little reputation that I may ever obtain, on the opinion, that conveyance inside the tunnels of pneumatic railways will, eventually, supersede conveyance in the open air, because it will be cheaper (in first cost, as well as current expenses), quicker, and safer while, in contrast with common railways and locomotive engines, I can show that it will be several times cheaper, in point of both first cost and current expenses; incomparably safer; and so much quicker, that few as are the hours required for conveying us a hundred miles by means of them, the public will, by and by (ere long, indeed) say as to those few hours, what Lear's daughters said of his knights.

I appeal to time as the umpire on these points; though one of my reasons for holding this opinion may be inferred from the table which follows the subjoined paragraphs, which I ask the favour of your inserting; which table was published in the same paper in which those paragraphs appeared, and applies to all carriages in the open air, be the means of traction what they may.

Wishing success to Messrs. Clegg and Samuda's application of the principle to the Dublin and Kingstown railway as interestedly as, and-for reasons which are stated in the second scene of the third act of King Henry the Fourth-m more anxiously than, they themselves desire it. I have the honour to be,

Sir, your most obedient servant, JOHN VALLANCE. 16, Brook-street, West-square, 25th July 1842.

THE OBJECTION TO TRAVELLING INSIDE PNEUMATIC RAILWAY TUNNELS CONSIDERED.

Nor would the objection which, it may be imagined, must arise from the want of daylight in the tunnel, prove an objection in point of fact. For, but that the advantages which would arise from saving expense in the purchase of ground, and obviation of objections on the part of landowners prevent it, I could window-light the tunnel throughout its whole length: that which I constructed at Brighton having light admitted into it through windows of common thin glass; strong plate glass not being required. Indeed, so far as relates to possibility, the upper half of the tunnel might be one continued window (like the top of a greenhouse) throughout its whole length. But as, even if this were done, artificial light must be had for sixteen hours out of the twenty-four in winter; as the tunnel might be gas-lighted throughout its whole length; or as, instead of thus wasting light unnecessarily, each carriage might carry lights before and behind, the objection that the tunnel being underground would render it dark as midnight, is no more a serious objection than it would be, were the Thames Tunnel finished, that it would be better to cross the river by London Bridge than through that tunnel, because on the bridge you would have natural, while in the tunnel you must have artificial light. It is true that there could be no "view of the country" by this method of travelling. But as the object of it is the perfection of conveyance, in the three particulars of safety, expedition, and economy; as even the comparatively low rates attained on the Liverpool and Manchester Railway prevent objects that are by the road-side being distinctly seen, owing to the velocity with which the passengers are whirled by them; and as the much greater velocity at which conveyance may be effected in the tunnel would render any attempt to look on what was

passed productive of the effects experienced by a child who looks on the ground while leaning out of the window of a coach, no real loss as relates to "seeing the country," would result from transmission taking place inside, instead of outside the tunnel: though even if it should, it might be submitted to when economy of both time and money, and complete obviation of the dangers attendant on breaking down, being overturned, run away with, or driven against anything, became the equivalents.

Have we occasion to travel to Edinburgh by the mail, we unrepiningly submit to the inconvenience of passing two nights (thirtytwo hours in mid-winter) not only in total darkness, but also "cabinned, cribbed, confined," to a degree which prevents us even from "changing a leg," except by previous arrangement with our opposite fellow-passenger. But when it is proposed that we shall go in vehicles which, in addition to being as large and commodious as the cabins of many steam-vessels, will be as much shorter a time in going, as they are larger and more convenient than the inside of mailcoaches, and in which the most brilliant light may be enjoyed, we proclaim it to be "impossible" to consent to go by such vehicles, because they would move inside a tunnel: not considering that this very circumstance of being inside said tunnel, would as certainly secure us from being overturned, driven against anything, run away with, breaking down, or any other of the dangers to which turnpike-road or railway travelling is liable, as it would give us the ease, comfort, and accommodation of the cabin of a steamvessel, instead of the privations and endurances experienced in mail-coaches. truly as-nay more truly than Hotspur says, "But I tell you, my lord fool, out of this nettle danger, we pluck this flower safety," may I say, from this bugbear darkness would spring perfect immunity from danger.

As

Sir,-When the recent alteration took place in our postage, of charging letters by weight, a great deal of ingenuity was exercised in devising balances suitable for that particular purpose; several of these, displaying considerable skill and fitness in their arrangement, were fully described at the time in your pages.

The late "sovereign panic" has shown, however, that numerous as were the dabblers in balances on the former occasion, the subject is not yet exhausted: several novelties having been produced, capable of satisfactorily indicating the minute deficiencies which wear and tear have produced on the gold coins of the country. I say minute, because the actual average deficiency is only from a grain to a grain and a half; each grain being worth 2d. Stories have been rife about the plugging" and "sweating" of our coins by Jews" and "foreigners," but with very little foundation; the deficiencies would seem really to arise from the fair wear of continuous circulation.

Many of your readers are doubtless

familiar with the "Adventures of a Guinea," but could one of our poor debased and degraded sovereigns describe the millions of rapid "changes" which it had undergone, between its issue from the Mint in the reign of the Sailor King, or the fourth of the Georges, or it may be, in that of his venerable predecessor, the wonder would be, that the actual deficiency should in so many cases be under half a grain.

The policy of throwing the loss on the holders of the deteriorated currency may be questioned: not so much as regards the actual deficiency, which is but small, but as opening the door to the most extensive frauds arising in some cases from sheer ignorance, in others from absolute knavery, by which the poorer classes have been robbed of from ninepence to fifteen pence, and even more in the pound.*

A larger portion of the mischief was incurred before there was any possibility

In some few instances as much as half-a-crown, and even three and sixpence have been deducted from a light sovereign."

"