that represented the numbers alive at the end of every annual stage of human life, from birth to the highest age attained. These threads were fixed by small sharp-pointed nails inserted in the wood.

A vertical sliding bar, or T, divided also into 10,000 parts, was then adjusted to a horizontal groove at the bottom of the triangle, and the instrument was set to work, in order to furnish life-annuity tables.

In constructing these on Barrett's system (improved by Mr Griffith Davies), all that is required in a single life table is, to multiply the numbers alive at the end of each year (starting with, say 10,000 just born) by the value of L.1 to be received at the end of 1, 2, 3, &c., years, until the highest age of the table is reached. Thus the number alive at the end of the first year is multiplied by the value of L.1 to be received 1 year hence at, say, 3 per cent. per annum, and the product is tabulated. The number alive at the end of the second year is multiplied by the value of L.1 to be received at the end of 2 years, and the product is also tabulated above the preceding result, and so on till the end, or longest life. The products are then summed up and added, one to the other, at every stage, beginning at the highest placed product and proceeding downwards. The sums thus obtained, called commutation values, only require to be divided by the products immediately below each sum, in order to obtain the value of an annuity for every age. Thus, if we take the lowest or initial number, 10,000, and divide by it the sum of all the products above it, we have a result which is made up of-1st, the value of L.1 payable 1 year hence, provided a child newly born survive one year; 2d, the value of L.1 payable 2 years hence, provided the child survive 2 years; 3d., the value of L.1, payable 3 years hence, provided the child survive 3 years, and so on to the highest age; in short, we have the value of an annuity on the life of a child just born, at the required rate of interest, and similar reasoning will show that the value of an annuity on the life of a child of 2 years, or any higher age, would be furnished with equal facility.

With reference to the action of the instrument; in order to have the products required, it is only necessary to shift the vertical sliding bar, or T, to the figure on the horizontal side

of the triangle indicating the value of L.1 to be received 1 year hence, and the point of intersection of the T with the line drawn from zero, at the left-hand corner, to the number alive at 1 year of age, on the perpendicular side, is the product required, and can be read off from the T and tabulated. By moving the T forward to the number representing the value of L.1 to be received 2 years hence, we get, at the point of intersection of the line drawn from zero to the number alive at the end of the second year, the required product, which can be similarly tabulated, and so on to the end of the table.

It was, however, found that the threads got, in the course of time, relaxed, and the products failed in the required accuracy.

The triangle was therefore filled up by seasoned wood, and the wood covered with paper. Lines were then ruled upon its surface similar to the threads, and the same results derived. The instrument is shown in the figure attached. The black lines are drawn from zero (on the left) to the vertical side up to the numbers (commencing at the top) of females alive, out of 97 newly born, at the end of the 1st, 2d, 3d, 4th, 5th, 6th, and 7th years from birth, according to the Government Tables, and the first 7 numbers to which the vertical dotted lines are drawn, on the horizontal side (reckoning from the right), are the values of L.1 due 1, 2, 3, 4, 5, 6, and 7 years hence, at 3 per cent. per annum, compound interest. The points of intersection of the 7 black lines drawn from zero with these dotted lines,-read on the vertical side of the triangle (as the action of the sliding bar, or T, cannot be represented in the drawing),-give the products of the respective numbers, viz., 82, 78, 75, 73, 69, 66, and 63. On adding these together, and dividing the sum by the number newly born, 97 (or more accurately 97.5), we have the value of an annuity of L.1, payable for 7 years upon the life of a female infant newly born, at 3 per cent. compound interest, by the said tables, or 5.189 pounds, while the value rigidly derived, by the ordinary mode, is 5.188, making a difference of only 001, or one farthing.

In making the calculations for joint lives, it was perceived

* See "The Government Annuity Tables," by the author. London, Groombridge and Sons, 1859, p. 763. 2 D

VOL. V.

that, in place of one value (or product) being obtained by a movement of the T, one hundred values could be got, or, in

fact, the values corresponding with the whole number of ages in the table. For, having obtained the products for a single life, all that is required to form a table of products for two joint lives is, to multiply the products already obtained for a single life by the numbers alive. Thus, for the joint life table of two equal ages, each product from age 1 to age 101, or end of the table, for a single life, would fall to be multiplied by the numbers alive at the end of 1 year, 2 years, 3 years, and so on to the end, in manner following, viz., the product for age 1 to be multiplied by the number alive at the end of 1 year; the product for age 2 to be multiplied by the number alive at the end of 2 years, and so on. By similar reasoning

the numbers required to be multiplied for the joint-life table, difference of age one, would be the product at age 1, for a single life, multiplied by the number alive at the end of 2 years, the product at age 2 multiplied by the number alive at the end of 3 years, and so on.

Now, in treating of the construction of a complete set of annuity tables for females, at, say 3 per cent. interest, as a whole, it is evident that the set may be constructed thus:

By fixing the T at the product of the number alive at 1 year, multiplied by the value of L.1 to be received 1 year hence, we have all the lines representing the numbers alive, from birth to the end of the table, cut at the points that give the multiplicates (or the results of the products multiplied by the numbers alive) which, being tabulated (or written down) horizontally, beginning with the youngest age on the left and proceeding to the right, forms the first step of the work.

By moving the T to the product of the number alive at 2 years multiplied by the value of L.1 to be received 2 years hence, we have the products required to form the second horizontal line of values, to be placed above the first line, commencing, as before, at the left hand and proceeding to the right. Continuing the process thus, up to the highest age, we have, arranged in vertical columns, the products which, on being summed up, give the commutation values for each table, from difference of age 0 to difference of age 100. If we require the value of an annuity for any particular ages, we find the same as before, by dividing the sum of the products above the required ages by the product of the numbers alive at the required ages multiplied by the value of L.1, to be received so many years hence as the age of the older extends to, a product already in the table, and being that immediately below the sum of the column.

Thus, without the aid of logarithms, and in the absence of multiplication, we obtain the products and commutation values which, in the ordinary way, are procured with great labour, although by the aid of logarithms.

The instrument will, if ruled accurately, give the multiplicates correct to 4 figures, and these will suffice to furnish the values of annuities correct to 3 decimal places besides the in

tegers, which is more than enough for every practical purpose.

So far as my experience goes, the instrument used by me. gives the values correctly to such an extent only that the values of the annuities for the youngest ages differ from the true values to the extent of 004, equivalent to one penny; but I have no doubt that, with more accurate ruling, this difference would disappear.

As the commutation form does not involve the giving of the actual annuities and assurances, the instrument furnishes, with the aid of simple addition, all the figures required; because, although the numbers alive only have been treated of, which give the annuities, the system is equally applicable to the decrements, or numbers dying in each year, from which, in a direct form, the commutation values are derived that give the values of life assurances.

By the process described, 400 values (or products) can with great accuracy be tabulated per hour-one party dictating, and another writing down the numbers; and, with a little practice, the number of values furnished per hour may be increased to 600.

The great labour experienced in forming life-annuity tables led me to consider as to a mechanical means of reducing such labour; and the instrument described, which was constructed by Messrs Alex. Adie and Son of this city, under my direction, although it has not yet been used in the actual construction of life-annuity tables, appears to me capable of accomplishing the desired result with comparative ease, and accurate to a sufficient number of decimals.

On the Construction of Iron Ships. By Mr THOMAS SHEDDON.*

Among the numerous shipwrecks that occurred towards the close of the past year, there were two which have called forth much remark, not only from the loss of life and property that attended them, but also from their being the wrecks of large iron sea-going steamers, and affording us some means of

* Read before the Society 27th February 1860.