Annuities have been treated of in the body of the work, but in such a manner as to render it necessary, that so useful a branch of knowledge should form the subject of an entirely new article in this place.

The doctrine of Compound Interest and Annuities—certain, is too simple ever to have occupied much of the attention of Mathematicians; inquiries into the values of interests dependent upon the continuance or the failure of human life, being more interesting and difficult, have occupied them more, but yet not so much as their importance would seem to demand; the discoveries, both in pure Mathematics and Physics, especially those of Newton, which distinguished the close of the seventeenth century, having provided them with ample employment of a more interesting kind, ever since the subjects of this article were submitted to calculation.

Pascal, Huygens, and Huygens, by laying the foundation of the doctrine of Probabilities about the middle of that century, first opened the way to the solution of problems of this kind. The earliest mathematical publication on Probabilities, the little tract of Huygens, De ratiociniis in ludo aleae, appeared in 1658; and in 1671, his celebrated countryman, John De Witt, published a treatise on Life-annuities, in Dutch. (Montucla, Hist. des Math. Tom. III. p. 407.) This, however, appears to have been very little-known, or read, and to have had no sensible influence on the subsequent progress of the science; the origin of which may be properly dated from the publication of Dr Halley's paper on the subject, in the Philosophical Transactions for the year 1693 (No. 196.) That celebrated Mathematician there first gave a table of mortality, which he had constructed from observations made at Breslaw, and showed how the probabilities of life and death, and the values of annuities and assurances on lives, might be determined by such tables; which, he informs us, had, till then, been only done by an imaginary valuation. Besides his algebraical reasonings, he illustrated the subject by the properties of parallelograms, and parallelopipeds; there are, perhaps, no other mathematical inquiries, in the prosecution of which, algebra is entitled to so decided a preference to the elementary geometry as in these, and this example of the application of geometry has not been followed by any of the succeeding writers.

In the year 1724, Mr De Moivre published the first edition of his tract, entitled, Annuities on Lives. In order to shorten the calculation of the values of such annuities, he assumed the annual decrements of life to be equal; that is, that out of a given number of persons living at any age, an equal number dies every year until they are all extinct; and, upon that hypothesis, he gave a general theorem, by which the values of annuities on single lives might be easily determined: this approximation, when the utmost limit of life was supposed to be 86 years, agreed very well with the true values between 30 and 70 years of age, as deduced from Dr Halley's table, and the method was of great use at the time; as no tables of the true values of annuities had then been calculated, except a very contracted one inserted by Dr Halley in the paper mentioned above. But, upon the whole, this hypothesis of De Moivre has probably contributed to retard the progress of the science, by turning the attention of Mathematicians from the investigation of the true law of mortality, and the best methods of constructing tables of the real values of annuities.

The same distinguished Analyst also endeavoured to approximate the values of joint lives; but it has since been found, that the formulae he gave for that purpose are too incorrect for use. Mr Thomas Simpson published his Doctrine of Annuities and Reversions in the year 1742, in which the subject is treated in a manner much more general and perspicuous than it had been previously; his formulae are adapted to any table of mortality, and, in the 7th corollary to his first problem, he gave the theorem demonstrated in the 149th number of this article, to which we owe all the best tables of the values of life-annuities that have since been published.

In the same work, he also gave a table of mortality deduced from the London observations, and four others calculated from it, of the values of annuities on lives, each at three rates of interest; the first for single lives, the three others for two and three equal joint lives, and for the longest of two or of three lives.

These were the first tables of the values of joint lives that had been calculated; for although Dr Halley had shown, half a century before, how such tables might be computed, and had taken considerable pains to facilitate the work; the necessary calculations, by the known methods, previous to the publication of Mr Simpson's Treatise, were so very labo- History, rious, that no one had had the courage to undertake them. And, unfortunately, the mortality, according to the London table, was so much above the common average, that the values of annuities in Mr Simpson's tables were much too small for general use.

In the year 1746, M. Deparcieux published his Essai sur les probabilités de la durée de la vie humaine, in which he gave several valuable tables of mortality deduced from the mortuary registers of different religious houses, and from the lists of the Nominees in the French Tontines; also a table of the values of annuities on single lives, at three rates of interest, calculated from his table of mortality for the tontine annuitants. These tables were a great acquisition to the science, as, before their publication, there were only two extant that gave tolerably exact representations of the true law of mortality:—Dr Halley's for Breslaw, and one constructed but a short time before by M. Kerseboom, principally from registers of Dutch annuitants. Those of M. Deparcieux for the Monks and Nuns, were the first ever constructed for the two sexes separately; and by them, the greater longevity of females was made evident.

The work commences with an algebraical theory of Annuities-certain; but the principal essay, On the Probabilities of the Duration of Human Life, is perfectly intelligible to those who have not studied Mathematics; it is written with great judgment and perspicuity, but contains very little more than the explanation of the construction of his tables, some of which relate to Tontines; and he did not avail himself to the extent he might have done, of the excellent tract of Thomas Simpson.

This work, however, appears to have been more read upon the Continent, and to have contributed more to the diffusion of this kind of information there, than all the other writings on the subject. The article Rentes viagères in the Encyclopédie, is acknowledged to have been taken entirely from it, as was also the article Vie, durée de la; and these are proofs, among many others that might be produced, how little M. D'Alembert and the principal Mathematicians, his contemporaries, attended to the subject.

In the year 1752, Mr. Simpson published, in his Select Exercises, a supplement to his Doctrine of Annuities; wherein he gave new tables of the values of annuities on two joint lives, and on the survivor of two lives, much more copious than those he had inserted in the principal work; but these also were calculated from his London table of mortality.

The celebrated Euler, in a paper inserted in the Memoirs of the Royal Academy of Sciences at Berlin for the year 1760, gave a formula by which the value of an annuity on a single life of any age, may be derived from that of an annuity on a life one year older; which formula was included in that given by Mr Simpson eighteen years before, for effecting the same purpose in the case of any number of joint lives; and by this compendious method, M. Euler calculated a table of the values of single lives from M. Kerseboom's table of mortality.

The first edition of Dr Price's Observations on Reversionary Payments was published in 1769; and its chief object was, to give information to persons desirous of forming themselves into societies for the purpose of making provision for themselves in old age, or for their widows. When tables of the values of single lives, and of two joint lives are given, the methods of determining the terms on which such provisions can be made with safety to all the parties concerned, are very simple, and were, at that time, well understood in theory, by the Mathematicians who had studied the subject; but, for want of the requisite tables, the algebraical formulæ had, till then, been of little practical utility.

In the prosecution of this laudable design, Dr Price was obliged to have recourse to approximations. He informs us, that by following M. De Moivre too implicitly in his rules for determining the value of two joint lives, he was led into difficulties which convinced him that they were not only useless, but dangerous; he therefore calculated a table of these values upon M. De Moivre's hypothesis of the decrements of life being equal, and its utmost limit 86 years, from a correct formula given by Mr Simpson in his Doctrine of Annuities (Cor. 5, Prob. I); by this, and a table of the values of single lives, calculated by Mr Dodson on M. De Moivre's hypothesis, he was enabled to give answers tolerably near the truth, to some of the most interesting questions of this kind, and to show that the plans of several of the societies then recently established, were quite inadequate; and instead of the benefits they promised, could only, in the end, produce disappointment and distress, unless they either dissolved or reformed themselves.

The work also contained instructive dissertations on the probabilities and expectations of life, and on the mean duration of marriage and of widowhood; besides accounts of some of the principal societies which had then been formed for the benefit of old age, and of widows; with observations on the method of forming tables of mortality for towns, and two new tables of that kind, constructed from registers kept at Norwich and Northampton. Mr Morgan's Doctrine of Annuities and Assurances was published in 1779, containing tables of the values of single lives, of two equal joint lives, and of two lives differing in age by 60 years, calculated from the Northampton table of mortality. And in the same year, M. De Saint-Cyran published his Calcul des Rentes viagères sur une et sur plusieurs têtes, wherein the valuation of annuities on lives is treated algebraically, but in a manner much inferior in all respects to that of Mr Simpson; and six tables are given of the values of annuities,—on single lives, on the survivor of two lives, and on the last survivor of three, calculated from M. Kerseboom's table of mortality. Although the values in the cases of two, and of three lives, were only determined by approximation, these tables were, just then, a valuable acquisition to the science; but their use was entirely superseded only four years after, by the publication of others much more valuable.

The fourth edition of Dr Price's Observations on Reversionary Payments appeared in 1783. One of the best effects of the preceding editions on the pro- gress of the science, had been, to direct the public attention to these inquiries, by showing their important uses in the affairs of life; and to procure the requisite data for forming tables of mortality, that should illustrate the laws according to which human life wastes under different circumstances, by exciting the curiosity of intelligent men who had the necessary leisure and means of information. The ingenious author had, accordingly, been furnished with the necessary abstracts of mortuary registers which had been kept with these views, by Dr Haygarth at Chester, Dr Aikin at Warrington, and the Rev. Mr Gor- such at Holy-Cross, near Shrewsbury, since the publication of the first edition; also by Mr Wargentin, with the mean numbers both of the living, and the annual deaths in all Sweden and Finland, for twenty-one successive years, in all of which the sexes were distinguished; and from these data, he constructed tables of mortality that threw great light on the subject. He also inserted in this edition, an improved table of mortality for Northampton; and, what had been so long wanted, a complete set of tables of the values of annuities on single lives, at six rates of interest, and on two joint lives at four, all calculated from the new Northampton table. The combinations of joint lives were sufficiently numerous to admit of all the values not included being easily interpolated. Besides these, he also gave tables of the values of annuities on single lives from the Swedish observations, both with and without distinction of the sexes, and on two joint lives without that distinction.

The values given in these tables are too low for the general average of lives, at all ages under 60; but in the treatise of Mr Baron Masceres on the Principles of the Doctrine of Life-Annuities, which was published in the same year (1783), others were given, calculated from the table of mortality which M. Deparcieux constructed from the lists of the Nomines in the French Tontines. The tables for single lives are calculated at twelve different rates of interest from 2 to 10 per cent.; but those for joint lives, only at 3½ and 4½ per cent.; and the combinations they include are only those of ages that are equal, or that differ by 5 or 10 years, and the multiples of 10.

There is reason to believe that the values in these tables, at all ages under 75 or 80 years, are nearer the truth, for the average of this country, than any others then extant; but certainly for the average of lives on which annuities and reversions depend. After that period of life, however, they are too small; and, in most cases, it is difficult to derive the values of joint lives from them with sufficient accuracy, on account of the contracted scale they have been calculated upon.

It was not Dr Price's object to deliver the elements of the science systematically; but he treated most parts of it with great judgment, enriched it with a vast collection of valuable facts and observations, and corrected several errors into which some of the most eminent writers upon it had fallen. The mathematical demonstrations (which are given in the notes) are much inferior to the rest of the work.

The values of reversionary sums and annuities, which depend upon some of the lives involved failing according to assigned orders of precedence, had been approximated by Mr Simpson in his Select Exercises; and by Mr Morgan in his Doctrine of Annuities; but the latter gentleman first gave accurate solutions of problems of this kind, in the Philosophical Transactions for the years 1788, 1789, 1791, 1794, and 1800.

Except by the solution of these problems, the science had not been materially advanced, during a period of more than thirty years that had elapsed since the appearance of the fourth edition of Dr Price's work, when Mr Milne published his Treatise on the Valuation of Annuities and Assurances, on Lives and Survivorships, in the beginning of last year (1815).

The work consists of two volumes; the first is mathematical, the second entirely popular, except the notes, and a few of the tables. The algebraical part of this article is merely a short abstract of the first volume, and may serve as a specimen of the manner in which the subject has been treated there; but the construction of tables of mortality, which forms the subject of the third chapter, has not been noticed here; neither is the valuation of reversionary sums or annuities depending upon assigned orders of survivorship, treated in the present article; and these are parts of the work, which will not be found the least interesting to mathematicians.

The second volume contains upwards of fifty new tables, with a few others that had been published before, but have been reprinted either on account of their value, or scarcity, or both. Four of the new ones are tables of mortality constructed by the author, from registers kept at Carlisle and Montpellier, and in all Sweden and Finland; since the period of the observations Dr Price made use of; the sexes are distinguished in the tables for Sweden and Montpellier, but not in that for Carlisle. This last is the only table, besides those for Sweden and Finland, that has been formed from the necessary data,—enumerations of the living, as well as registers of the deaths in every interval of age.

Twenty-one of these tables, being the seventeenth to the thirty-seventh inclusive, in the collection at the end of the work, render it easy to apply the algebraical formulae to practical purposes, and numerous examples of such applications are given. They have all been calculated from the Carlisle table of mortality; those of the values of life-annuities on the same extensive scale, with those which Dr Price derived from the Northampton table. It is the author's opinion that the values of interests dependent upon the continuance or the failure of life, may be derived from them more correctly than from any others extant; and he has taken considerable pains to assist his readers in judging of this for themselves.

Besides the tables, the principal contents of the second volume, are explanations of their construction and uses; many of them relate to the progress of population,—the comparative mortality of different diseases—of different seasons,—and of the two sexes at every age—the proportion of the sexes at birth—and that of the born alive to the still-born of each sex. It will be found that the author has been furnished with facts and observations of great value, and that he has endeavoured to present the information they afford, in the forms best calculated for the further prosecution of these inquiries.

In treating of annuities, we think that it may be useful in a work of this kind, to address ourselves as well to those readers who have not, as to those who have, an acquaintance with Algebra; and we shall, accordingly, divide what follows into two Parts, corresponding to these two views of the subject.

PART I.

We shall, in this Part, demonstrate all that is most useful and important in the doctrine of Annuities and Assurances on lives, without using Algebra, or introducing the idea of probability; but the reader is, of course, supposed to understand common Arithmetic. In the first 30 numbers of this Part, Compound Interest and Annuities-certain are treated of; from the 31st to the 76th, the doctrine of Annuities on Lives is delivered; and that of Assurances on Lives, from thence to the 108th, where the popular view terminates.

What is demonstrated in this Part, will be sufficient to give the reader clear and scientific views of the subjects treated; and, with the assistance of the necessary tables, will enable him to solve the more common and simple problems respecting the values of Annuities and Assurances. He will also understand clearly the general principles on which problems of greater difficulty are resolved; but these he cannot undertake with propriety, when the object is, to make a fair valuation of any claims or interests, with a view to an equitable distribution of property, unless he has studied the subject carefully, with the assistance of Algebra; for intricate problems of this kind can hardly be solved without it; and those who are not much exercised in such inquiries, often think they have arrived at a complete solution, when they have overlooked some circumstance or event, or some possible combination of events or circumstances, which materially affect the value sought. Eminent Mathematicians have, in this way, fallen into considerable errors, and it can hardly be doubted, that those who are not mathematicians, must (ceteris paribus) be more liable to them.

I. ON ANNUITIES-CERTAIN.

No. 1. When the rate is 5 per cent., L.1 improved at simple interest during one year, will amount to L.1·05; which, improved in the same manner during the second year, will be augmented in the same ratio of 1 to 1·05; the amount then, will therefore be $1 \times 1\cdot05$, or $(1\cdot05)^2 = 1\cdot1025$.

In the same manner it appears, that this last amount, improved at interest during the third year, will be increased to $(1\cdot05)^3 = 1\cdot157625$; at the end of the fourth year, it will be $(1\cdot05)^4$; at the end of the fifth $(1\cdot05)^5$, and so on; the amount at the end of any number of years being always determined, by raising the number which expresses the amount at the end of the first year, to the power of which the exponent is the number of years. So that when the rate of interest is 5 per cent., L.1 improved at compound interest, will, in seven years, amount to $(1\cdot05)^7$, and in 21 years, to $(1\cdot05)^{21}$.

But if the rate of interest were only 3 per cent., these amounts would only be $(1\cdot03)^7$, and $(1\cdot03)^{21}$ respectively.

2. The present value of L.1 to be received certainly at the end of any assigned term, is such a less sum, as, being improved at compound interest during the term, will just amount to one pound. It must therefore be less than L.1, in the same ratio as L.1 is less than its amount in that time; but in three years, at 5 per cent., L.1 will amount to $(1\cdot05)^3$.

And $(1\cdot05)^3 : 1 :: 1 : \frac{1}{(1\cdot05)^3}$, so that $\frac{1}{(1\cdot05)^3} = \frac{1}{1\cdot157625} = 0\cdot863838$ is the present value of L.1 to be received at the expiration of three years.

In the same manner it appears that, at 4 per cent. interest, the present value of L.1 to be received at the end of a year, is $\frac{1}{1\cdot04} = 0\cdot961538$; and if it were not to be received until the expiration of 21 years, its present value would be $\frac{1}{(1\cdot04)^{21}} = (0\cdot961538)^{21} = 0\cdot438834$.

Hence it appears, that if unity be divided by the amount of L.1, improved at compound interest during any number of years, the quotient will be the present value of L.1 to be received at the expiration of the term: which may also be obtained by raising the number which expresses the present value of L.1 receivable at the expiration of a year, to the power of which the exponent is the number of years in the term.

3. When a certain sum of money is receivable annually, it is called an Annuity, and its quantum is expressed by saying it is an annuity of so much; thus, according as the annual payment is L.1, L.10, or L.100; it is called an annuity of L.1, of L.10, or of L.100.

4. When the annual payment does not depend upon any contingent event, but is to be made certainly, either in perpetuity or during an assigned term, it is called an Annuity-certain.

5. In calculating the value of an annuity, the first payment is always considered to be made at the end of the first year from the time of the valuation, unless the contrary be expressly stated.

6. The whole number, and part or parts of one annual payment of an annuity, which all the future payments are worth in present money, is called the number of years purchase the annuity is worth; and, being the sum of the present values of all the future payments, is also the sum which, being put out and improved at compound interest, will just suffice for the payment of the annuity (2).

7. Hence it follows, that when the annuity is L.1, the number of years purchase and parts of a year, is the same as the number of pounds and parts of a pound in its present value.

And throughout this article, whenever the quantum of an annuity is not mentioned, it is to be understood to be L.1.

8. The sum of which the simple interest for one year is L.1, is evidently that which, being put out at interest, will just suffice for the payment of L.1 at the end of every year, without any augmentation or diminution of the principal; and, being equivalent to the title to L.1 per annum for ever, is called the value of the perpetuity, or the number of years purchase the perpetuity is worth.

But, while the rate remains the same, the annual interests produced by any two sums, are to each other as the principals which produce them; therefore, since $5 : 1 :: 100 : \frac{100}{5} = 20$, when the rate of interest is 5 per cent., the value of the perpetuity is 20 years purchase. In the same manner it appears, that according as the rate may be 3 or 6 per cent., the value of the perpetuity will be $\frac{100}{3} = 33\frac{1}{3}$, or $\frac{100}{6} = 16\frac{2}{3}$ years purchase; and may be found in every case, by dividing any sum by its interest for a year.

9. All the most common and useful questions in the doctrines of compound interest and annuities certain, may be easily resolved by means of the first four tables at the end of this article. Their construction may be explained by the following specimen, rate of interest 5 per cent.

| Table IV | Table III | Table I | Table II | |----------|-----------|---------|---------| | Amount of L.1 per annum | Amount of L.1 | Present value of L.1 per annum, received until | Present value of L.1 per annum, received until | | Term | until the expiration of the Term | Term | |

10. The calculation must begin with Table III., the first number in which should evidently be 1.05, the amount of L.1 improved at interest during one year; which, being multiplied by 1.05, the product is 1.05², the second number; this second number being multiplied by 1.05, the product is 1.05³, the amount at the end of three years. And so the calculation proceeds throughout the whole of the column; each number after the first, being the product of the multiplication of the preceding number, by the amount of L.1 in a year (1).

11. The number against any year in Table I. is found by dividing unity by the number against the same year in Table III. (2); thus, the number against the term of six years in Table I. is $\frac{1}{1.340096} = 0.746215$. All the numbers in that table after the first, may also be found, by multiplying that first number continually into itself (3).

12. The number against any year in Table II. being the sum of the numbers against that and all the preceding years in Table I.; is found by adding the number against that year in Table I. to the number against the preceding year in Table II.; thus, the number against 4 years in Table II., being

the sum of 0.822702 and 2.723248

is 3.545950.

13. If each payment of an annuity of L.1 be put out as it becomes due, and improved at compound interest during the remainder of the term, it is evident that at the expiration of the term, the payment then due will be but L.1, having received no improvement at interest. That received one year before will be augmented to the amount of L.1 in a-year; that received two years before will be augmented to the amount of L.1 in two years; that received three years before to the amount of L.1 in three years, and so on until the first payment, which will be augmented to the amount of L.1 in a term one year less than that of the annuity.

Hence, it is manifest, that the number against any year in Table IV. will be unity added to the sum of all those against the preceding years in Table III.

And, therefore, that the number against any year in Table IV. is the sum of those in Tables III. and IV. against the next preceding year.

Thus, the number against seven years in Table IV., being

the sum of 1.340096 and 6.801913

is 8.142009.

14. The method of construction is obviously the same at any other rate of interest.

15. All the amounts and values which are the objects of this inquiry, evidently depend upon the improvement of money at compound interest; it is, therefore, that the first, second, and fourth tables, all depend upon the third.

But every pound, and every part of a pound, when put out at interest, is improved in the same manner as any single pound considered separately. Whence, it is obvious, that while the term and the rate of interest remain the same, both the amount and the present value, either of any sum, or of any annuity, will be the same multiple, and part or parts of the amount or the present value found against the same term, and under the same rate of interest in these tables, as the sum or the annuity proposed is of L.1.

So that to find the amount or the present value of any sum or annuity for a given term and rate of interest, we have only to multiply the corresponding tabular value by the sum or the annuity proposed; the product will be the amount or the value sought, according as the case may be.

16. Example 1. To what sum will L. 100 amount, when improved at compound interest during 20 years? the rate of interest being 4 per cent. per annum.

By Table III., it appears, that L. 1 so improved, would, at the expiration of the term, amount to L. 2·191123, therefore L. 100 would amount to 100 times as much, that is, to L. 219·1123, or L. 219, 2s. 3d.

17. Ex. 2. What is the present value of L. 400, which is not to be received until the expiration of 14 years, when the rate of interest is 5 per cent.?

The present value of L. 1 to be received then, will be found by Table I. to be L. 0·505068 : L. 400 to be received at the same time, will therefore be worth, in present money, 400 times as much, or L. 202·0272, that is, L. 202, 0s. 6½d.

18. Ex. 3. Required the present value of an annuity of L. 50 for 21 years, when the rate of interest is 5 per cent.

Table II. shows the value of an annuity of L. 1 for the same term to be L. 12·8212 ; the required value must therefore be 50 times as much, or L. 641·06, that is, L. 641, 1s. 2½d.

19. Ex. 4. What will an annuity of L. 10, 10s. or L. 10·5, for thirty years, amount to, when each payment is put out as it becomes due, and improved at compound interest until the end of the term? The rate of interest being 4 per cent.

The amount of an annuity of L. 1 so improved, would be L. 560·84938, as appears by Table IV., the amount required will therefore be 10·5 times this, or L. 588·89185, that is, L. 588, 17s. 10d.

20. When the interval between the time of the purchase of an annuity and the first payment thereof, exceeds that which is interposed between each two immediately successive payments; such annuity is said to be deferred for a time equal to that excess, and to be entered upon at the expiration of that time.

21. If two persons, A and B, purchase an annuity between them, which A is to enter upon immediately, and to enjoy during a certain part of the term, and B, or his heirs, or assigns, for the remainder of it; the present value of B's interest will evidently be, the excess of the value of the annuity for the whole of the term from this time, above the value of the interest of A.

So that when the entrance on an annuity is deferred for a certain term, its present value will be the excess of the value of the annuity for the term of delay and continuance together, above the value of an equal annuity for the term of delay only.

22. Example 1. Required the value of a perpetual annuity of L. 120, which is not to be entered upon until the expiration of 14 years from this time, reckoning interest at 3 per cent.

The perpetuity, with immediate possession, would be worth 33½ years' purchase (8); and an annuity for the term of delay is worth 11·2961 (Table II).

From 33·3333 subtract 11·2961, and multiply the remainder 22·0372 by 120

the product 2644·464 = L. 2644, 9s. 3½d.

is the required value.

23. Ex. 2. Allowing interest at 5 per cent. what sum should be paid down now for the renewal of 14 years lapsed, in a lease for 21 years of an estate producing L. 300 per annum, clear of all deductions?

This is the price of an annuity for 14 years, to be entered upon 7 years hence; the term of delay, therefore, is 7 years, and that of the delay and continuance together 21 years.

By Table II. it appears, that the present value of an annuity for 21 years, is 12·8212 ; for 7 years, 5·7864 ; years' purchase.

Value of the deferred annuity, 7·0348 Multiply by 300

The product L. 2110·44, or L. 2110, 8s. 9½d. is the price required.

24. Hitherto we have proceeded upon the supposition of the annuity being payable, and the interest convertible into principal, which shall reproduce interest, only once a-year.

But annuities are generally payable half-yearly, and sometimes quarterly; and the same circumstances that render it desirable for an annuitant to receive his annual sum in equal half-yearly or quarterly portions, also give occasion to the interest of money being paid in the same manner.

But whatever has been advanced above, concerning the present value or the amount of an annuity, when both that and the interest of money were only payable once a-year, will evidently be true when applied to half the annuity, and half the interest paid twice as often, on the supposition of half-yearly payments; or to a quarter of the annuity, and a quarter of the interest, paid four times as often, when the payments are made quarterly.

25. Half-yearly payments are, however, by far the most common, and these four tables will also enable us to answer the most useful questions concerning them.

For we have only to extract the present value, or the amount, from the table, against twice the number of years in the term, at half the annual rate of interest, and, in the case of an annuity, to multiply the number so extracted, by half the annuity proposed.

26. Ex. 1. To what sum will L. 100 amount in 20 years, when the interest at the rate of 4 per cent. per annum, is convertible into principal half-yearly?

This being the amount in 40 half years at 2 per cent. interest for every half year, will be the same as the amount in 40 years at 2 per cent. per annum, which, by Table III. will be found to be 220·804, or L. 220, 16s. 1d.; and is only L. 1, 13s. 10d. more than it would amount to if the interest were not convertible more than once a-year (16). 27. Ex. 2. What is the present value of an annuity of L. 50 for 21 years, receivable in equal half-yearly payments, when money yields an interest of 2½ per cent. every half year?

By Table II. it appears, that an annuity of L. 1 for 42 years, when the interest of money is 2½ per cent. per annum, will be worth L. 25:8206 (25); 25 times this sum, or L. 645, 10s. 3½d. is therefore, the required value, and exceeds the value when the interest and the annuity are only payable once a-year by L. 4, 9s. 1d. (18).

28. The excess of an annuity-certain above the interest of the purchase-money, is the sum which, being put out at the time of each payment becoming due, and improved at compound interest until the expiration of the term, will just amount to the purchase-money originally paid.

But, while everything else remains the same, the longer the term of the annuity is, the less must its excess above the interest of the purchase-money be, because a less annuity will suffice for raising the same sum within the term. Therefore, the proportion of that excess to the annual interest of the purchase-money, continually diminishes as the term is extended; and when the annuity is a perpetuity, there is no such excess (8).

29. The reason why the value of an annuity is increased by that and the interest being both payable more than once in the year, is, that the grantor loses, and the purchaser gains, the interest produced by that part of each payment, which is in excess above the interest then due upon the purchase-money, from the time of such payment being made, until the expiration of the year.

Hence it is obvious, that the less this excess is, that is, the longer the term of the annuity is (28), the less must the increase of value be.

And when the annuity is a perpetuity, its value will be the same, whether it and the interest of money be both payable several times in the year, or once only.

30. When the annuity is not payable at the same intervals at which the interest is convertible into principal, its value will depend upon the frequencies both of payment and conversion; but its investigation without algebra, would be too long, and of too little use, to be worth prosecuting here.

II. OF ANNUITIES ON LIVES.

31. When the payment of an annuity depends upon the existence of some life or lives, it is called a Life-annuity.

32. The values of such annuities are calculated by means of tables of mortality, which show, out of a considerable number of individuals born, how many upon an average have lived to complete each year of their age; and, consequently, what proportion of those who attained to any one age, have survived any greater age.

The fifth Table at the end of this article is one of that kind, which has been taken from Mr Milne's Treatise on Annuities, and was constructed from accurate observations made at Carlisle by Dr Heysham, during a period of 9 years, ending with 1787.

33. By this table it appears, that during the period in which these observations were made; out of 10,000 children born, 3203 died under 5 years of age, and the remaining 6797 completed their fifth year. Also, that out of 6797 children who attained to 5 years of age, 6460 survived their 10th year.

But the mortality under 10 years of age, has been greatly reduced since then, by the practice of vaccination.

This table also shows, that of 6460 individuals who attained to 10 years of age, 6047 survived 21. And that of 5075 who attained to 40, only 3643 survived their 60th year.

34. There is good reason to believe (as has been shown in another place), that the general law of mortality, that is, the average proportion of persons attaining to any one age, who survive any greater age, remains much the same now among the entire mass of the people throughout England, as it was found to be at Carlisle during the period of these observations; except among children under 10 years of age, as was noticed above (33).

If this be so, it will follow, that of 6460 children now 10 years of age, just 6047 will attain to 21; or rather, that if any great number be taken in several instances, this \( \frac{6047}{6460} \) will be the average proportion of them that will survive the period.

And if 6460 children were to be taken indiscriminately from the general mass of the population at 10 years of age, and an office or company were to engage to pay L. 1, eleven years hence, for each of them that might then be living; this engagement would be equivalent to that which should bind them to pay L. 6047 certainly, at the expiration of the term.

Therefore, the office, in order that it might neither gain nor lose by the engagement, should, upon entering into it, be paid for the whole, the present value of L. 6047; to be received at the expiration of 11 years; and for each life, the \( \frac{1}{6460} \)th part of it; that is, the \( \frac{6047}{6460} \)th part of the present value of L. 1 to be received then.

But when the rate of interest is 5 per cent. the present value of L. 1, to be received at the expiration of 11 years, is L. 0:584679; therefore, at that rate of interest, there should be paid for each life \( \frac{6047 \times 0:584679}{6460} = L. 0:5473 \).

And the present value of L. 100, to be received upon a life now 10 years of age attaining to 21, will be L. 54:73, or L. 54, 18s. 7d.

In the same manner it will be found, that reckoning interest at 4 per cent. the value would be L. 60, 16s. 1d.

35. This is the method of calculating the present values of endowments for children of given ages; and the values of annuities on lives may be computed in the same manner.

For, from the above reasoning it is manifest, that if the present value of L. 1, to be received certainly at the expiration of a given term, be multiplied by the number in the table of mortality against the age, greater than that of any proposed life by the number of years in the term, and the product be divided by the number in the same table, against the present age of that life; the quotient will be the present value of L.1, to be received at the expiration of the term, provided that the life survive it.

And if, in this manner, the value be determined of L.1, to be received upon any proposed life, surviving each of the years in its greatest possible continuance, according to the table of mortality adapted to it; that is, according to the Carlisle table, upon its surviving every age greater than its present, to that of 104 years inclusive; then, the sum of all these values will evidently be the present value of an annuity on the proposed life.

36. If 5642 lives at 30 years of age be proposed, and 5075 at the age of 40; since each of the 5642 younger lives may be combined with every one of the 5075 that are 10 years older, the number of different pairs, or different combinations of two lives differing in age by 10 years, that may be formed out of the proposed lives, is 5642 times 5075.

But at the expiration of 15 years, the survivors of the lives now 30 and 40 years of age, being then of the respective ages of 45 and 55, will be reduced to the numbers of 4727 and 4073 respectively; and the number of pairs, or combinations of two, differing in age by 10 years, that can be formed out of them, will be reduced from 5642 X 5075 to 4727 X 4073.

So that L.1 to be paid at the expiration of 15 years for each of these 5642 X 5075 pairs or combinations of two, now existing, which may survive the term, will be of the same value in present money, as 4727 times L.4073, to be received certainly at the same time.

Now let A be any one of these lives of 30 years of age, and B any one of those aged 40; and from what has been advanced it will be evident, that the present value of L.1 to be received upon the two lives in this particular combination jointly surviving the term, will be the same as that of the sum L.4727 X 4073 to be then received certainly.

But, when the rate of interest is 5 per cent, L.1 to be received certainly at the expiration of 15 years, is equivalent to L.0.481017 in present money (Table I).

Therefore, at that rate of interest, and according to the Carlisle table of mortality; the present value of L.1 to be received upon A and B now aged 30 and 40 years respectively, jointly surviving the term of 15 years, will be

\[ \frac{4727 \times 4073 \times L.0.481017}{5642 \times 5075} \]

37. Hence it is sufficiently evident, how the present value of L.1 to be received upon the same two lives jointly surviving any other year may be found. And if that value for each year from this time until the eldest life attain to the limit of the table of mortality be calculated, the sum of all these will be the present value of an annuity of L.1 dependent upon their joint continuance.

In this manner, it is obvious that the value of an annuity on the joint continuance of any other two lives might be determined.

38. If, besides the 5642 lives at 30 years of age, and the 5075 at 40 (mentioned in No. 36), there be also proposed 3643 at 60 years of age; each of these 3643 at 60, may be combined with every one of the 5642 X 5075 different combinations of a life of 30, with one of 40 years of age; and, therefore, out of these three classes of lives 5642 X 5075 X 3643 different combinations may be formed; each containing a life of 30 years of age; another of 40, and a third of 60.

But at the expiration of 15 years, the numbers of lives in these three classes will, according to the table of mortality, be reduced to 4727, 4073, and 1675 respectively; the respective ages of the survivors in the several classes being then 45, 55, and 75 years; and the number of different combinations of three lives (each of a different class from either of the other two), that can be formed out of them, will be reduced to 4727 X 4073 X 1675.

Hence, by reasoning as in No. 36, it will be found, that if A, B, and C be three such lives, now aged 30, 40, and 60 years, the present value of L.1 to be received upon these three jointly surviving the term of 15 years from this time, will be

\[ \frac{4727 \times 4073 \times 1675}{5642 \times 5075 \times 3643} \times L.0.481017 \]

interest being reckoned at 5 per cent.

Thus it is shown, how the present value of an annuity dependent upon the joint continuance of these three lives might be calculated, that being the sum of the present values thus determined, of the rents for all the years which, according to the table of mortality, the eldest life can survive.

39. But it is easy to see, that the same method of reasoning may be used in the case of four, five, or six lives, and so on without limit. Whence, this inference is obvious.

The present value of L.1, to be received at the expiration of a given term, provided that any given number of lives all survive it, may be found by multiplying the present value of L.1 to be received certainly at the end of the term, by the continual product of the numbers in the table of mortality against the ages greater respectively by the number of years in the term, than the ages of the lives proposed; and dividing the last result of these operations, by the continual product of the numbers in the table of mortality against the present ages of the proposed lives.

And by a series of similar operations, the present value of an annuity on the joint continuance of all these lives might be determined.

But it should be observed, that, in calculating the value of a life-annuity in this way, the denominator of the fractions expressing the values of the several years rents, that is, the divisor used in each of the operations, remains always the same; the division should, therefore, be left till the sum of the numerators is determined, and one operation of that kind will suffice.

40. Enough has been said to show that these methods of constructing tables of the values of annuities on lives are practicable, though excessively laborious, and, in fact, all the early tables of this kind were constructed in that manner. We proceed now to show how such tables may be calculated with much greater facility.

41. By the method of No. 34, it will be found that, reckoning interest at 5 per cent., the present value of L. 1 to be received at the expiration of a year, provided that a life, now 89 years of age, survived till then, is \( \frac{142 \times 0.952381}{181} \). But the age of that life will then be 90 years, and the proprietor of an annuity of L. 1 now depending upon it, will, in that event, receive his annual payment of L. 1 then due; therefore, if the value then of all the subsequent payments, that is, the value of an annuity on a life of 90 be 2:339 years' purchase, the present value of what the title to this annuity may produce to the proprietor, at the end of the year, will be the same as that of L. 3:339, to be received then, if the life be still subsisting, or \( \frac{142 \times 0.952381}{181} \times L. 3:339 = L. 2:495 \); which, therefore, will be the present value of an annuity of L. 1 on a life of 89 years of age. That is to say, an annuity on that life will now be worth 2:495 years' purchase (7).

42. In the same manner it appears generally, that, if unity be added to the number of years' purchase that an annuity on any life is worth, and the sum be multiplied by the present value of L. 1, to be received at the end of a year, provided that a life one year younger survive till then, the product will be the number of years' purchase an annuity on that younger life is worth in present money.

43. But according to the table of mortality, an annuity on the eldest life in it is worth nothing; therefore, the present value of L. 1 to be received at the end of a year, provided that the eldest life but one in the table survive till then, is the total present value of an annuity of L. 1 on that life. Which, being obtained, the value of an annuity on a life one year younger than it may be found by the preceding number; and so on for every younger life successively.

**EXEMPLIFICATION.**

| Age of Life | Value of an Annuity on that Life, increased by Unity. | Which, being multiplied by 0.952381, and the Product by | |-------------|------------------------------------------------------|-----------------------------------------------------| | 104 | 1:000 | \( \frac{1 \times 40}{3 \times 54} \) | | 103 | 1:317 | \( \frac{3 \times 54}{5 \times 75} \) | | 102 | 1:753 | \( \frac{5 \times 75}{7 \times 105} \) | | 101 | 2:192 | \( \frac{7 \times 105}{9 \times 142} \) | | 100 | 2:624 | \( \frac{9 \times 142}{11 \times 181} \) | | 99 | 3:045 | \( \frac{11 \times 181}{14 \times 232} \) | | 98 | 3:478 | | | 97 | 3:828 | | | 96 | 3:555 | | | 95 | 3:996 | |

44. Proceeding as in No. 36, it will be found, that at 5 per cent. interest, and according to the Carlisle table of mortality, the present value of L. 1 to be received at the expiration of a year, provided that a person now 89 years of age, and another now 99, be then living, is \( \frac{142 \times 9 \times L. 0.952381}{181 \times 11} \); therefore, if the present value of an annuity of L. 1 on the joint continuance of two lives, now aged 90 and 100 years respectively, be L. 0.950; by reasoning as in No. 41, it will be found that the present value of an annuity on the joint continuance of two lives, of the respective ages of 89 and 99 years, will be worth \( \frac{142 \times 9 \times 0.952381}{181 \times 11} \times 1.950 = 1.192 \) years' purchase.

45. In this manner, commencing with the two oldest lives in the table that differ in age by ten years, and proceeding according to No. 43, the values of annuities on all the other combinations of two lives of the same difference of age, may be determined.

The method is exemplified in the following specimen:

| Ages of two Lives | Value of an Annuity on their joint continuance, increased by Unity. | Which, being multiplied by 0.952381, and the Product by | Produces the value of an Annuity on the two joint Lives one year younger respectively, | |------------------|-------------------------------------------------------------------|-----------------------------------------------------|---------------------------------------------------------------------------------| | 94 & 104 | 1:000 | \( \frac{1 \times 40}{3 \times 54} \) | 0:235 | | 93 & 103 | 1:235 | \( \frac{3 \times 54}{5 \times 75} \) | 0:508 | | 92 & 102 | 1:508 | \( \frac{5 \times 75}{7 \times 105} \) | 0:733 | | 91 & 101 | 1:733 | \( \frac{7 \times 105}{9 \times 142} \) | 0:950 | | 90 & 100 | 1:950 | \( \frac{9 \times 142}{11 \times 181} \) | 1:192 | | 89 & 99 | 2:192 | \( \frac{11 \times 181}{14 \times 232} \) | 1:280 |

46. Hence, and by what has been advanced in the 39th number of this article, it is sufficiently evident, how a table may be computed of the values of annuities on the joint continuance of the lives in every combination of three, or any greater number; the differences between the ages of the lives in each combination remaining always the same in the same series of operations, while the calculation proceeds back from the combination in which the oldest life is the oldest in the table, to that in which the youngest is a child just born.

47. But, when there are more than two lives in each combination, the calculations are so very laborious, on account, principally, of the great number... of combinations, that no tables of that kind have yet been published, except three or four for three lives.

And, in the books that contain tables of the values of two joint lives, methods are given of approximating towards the values of such combinations of two and of three lives, as have not yet been calculated.

Therefore, assuming the values of annuities on single lives, and on the joint continuance of two or of three lives, to be given; we have next to show how the most useful problems respecting the values of any interests that depend upon the continuance or the failure of life, may be resolved by them.

48. Proposition 1. The value of an annuity on the survivor of two lives, A and B, is equal to the excess of the sum of the values of annuities on the two single lives, above the value of an annuity on their joint continuance.

49. Demonstration. If annuities on each of the two lives were granted to P, during their joint continuance, he would have two annuities; but if P were only to receive these upon condition that, during the joint lives of A and B, he should pay one annuity to Q; then, there would only remain one to be enjoyed by him, or his heirs or assigns, until the lives both of A and B were extinct; whence the truth of the proposition is manifest.

50. Prop. 2. The value of an annuity on the joint continuance of the two last survivors out of three lives, A, B, and C, is equal to the excess of the sum of the values of annuities on the three combinations of two lives (A with B, A with C, and B with C) that can be formed out of them, above twice the value of an annuity on the joint continuance of all the three lives.

51. Demon. If one annuity were granted to P on the joint continuance of the two lives A and B, another on the joint continuance of A and C, and a third, on the joint continuance of B and C; during the joint continuance of all the lives he would have three annuities.

But if he were only to receive these upon condition that he should pay two annuities to Q, during the joint continuance of all the three lives; then, there would only remain to himself one annuity during the joint existence of the last two survivors out of the three lives. And the truth of the proposition is manifest.

52. Prop. 3. The value of an annuity on the last survivor of three lives, A, B, and C, is equal to the excess of the sum of the values of annuities on each of the three single lives, together with the value of an annuity on the joint continuance of all the three, above the sum of the values of three other annuities; the first dependent upon the joint continuance of A and B, the second, on that of A and C, and the third, on B and C.

53. Demon. If annuities on each of the three single lives were granted to R; during the joint continuance of all the three, he would have three annuities, and from the time of the extinction of the first life that failed, till the extinction of the second, he would have two.

So that he would have two annuities during the joint existence of the two last survivors out of the three lives; and besides these, a third annuity during the joint continuance of all the three.

Therefore, if out of these, R were to pay one annuity to P during the joint continuance of the last two survivors out of the three lives, and another to Q during the joint continuance of all the three; he would only have left one annuity, which would be receivable during the life of the last survivor of the three.

But in the demonstration of the last proposition (51) it was shown, that the value of what he paid to P, would fall short of the sum of the values of annuities dependent respectively on the joint continuance of A and B, of A and C, and of B and C, by two annuities on the joint continuance of all the three lives. Whence it is evident, that the value of the annuities he paid both to P and Q, would fall short of the sum of these three values of joint lives, only by the value of one annuity on the joint continuance of all the three lives.

Wherefore, if from the sum of the values of all the three single lives, the sum of the values of the three combinations of two that can be formed out of them were taken; there would remain less than the value of an annuity on the last survivor, by that of an annuity on the joint continuance of the three lives.

But if, to the sum of the values of the three single lives A, B, and C, there be added that of an annuity on the joint continuance of the three, and from the sum of these four values, the sum of the values of the three combinations, A with B, A with C, and B with C be subtracted; then, the remainder will be the value of an annuity on the last survivor of the three lives. Which was to be demonstrated.

54. Prop. 4. Problem. The law of mortality and the values of single lives at all ages being given; to determine the present value of an annuity on any proposed life, deferred for any assigned term.

55. Solution. Find the present value of an annuity on a life older than the proposed, by the number of years during which the other annuity is deferred; multiply this by the present value of L.1 to be received upon the proposed life surviving the term, and the product will be the value sought.

56. Demon. Upon the proposed life surviving the term, the annuity dependent upon it will be worth the same sum, that an annuity on a life so much older is now worth; therefore, it is evident, that the deferred annuity is of the same present value as that sum to be received at the expiration of the term, provided the life survive it.

57. Corollary. In the same manner it appears, that the present value of an annuity on the joint continuance of any number of lives, deferred for a given term, may be found by multiplying the present value of an annuity on the joint continuance of the same number of lives, older respectively than the proposed, by the number of years in the term; by the present value of L.1 to be received, upon the proposed lives all surviving it.

58. A Temporary Annuity on any single life, or on the joint continuance of any number of lives, that is, an annuity which is to be paid during a certain term, provided that the single life or the other lives jointly subsist so long; together with an annuity on the same life or lives deferred for the same term, are evidently equivalent to an annuity on the whole duration of the same life or lives. So that the value of an annuity on any life or on the joint continuance of any number of lives for an assigned term, is equal to the excess of the value of an annuity on their whole duration, with immediate possession, above the value of an annuity on them deferred for the term.

59. Whatever has been advanced from No. 48. to 53. inclusive, respecting the values of annuities for the whole duration of the lives wherein they depend, will apply equally to those which are either deferred or temporary; and, therefore, to determine the value of any deferred or temporary annuity, dependent upon the last survivor of two or of three lives; or, upon the joint continuance of the last two survivors out of three lives; we have only to substitute temporary or deferred annuities, as the case may require, for annuities on the whole duration of the lives; and the result will, accordingly, be the value of a temporary or deferred annuity on the life of the last survivor, or on the joint lives of the two last survivors.

60. Prop. 5. A and B being any two proposed lives now in existence, the present value of an annuity to be payable only while A survives B, is equal to the excess of the value of an annuity on the life of A, above that of an annuity on the joint existence of both the lives.

61. Demons. If an annuity on the life of A, and to be entered upon immediately, were now granted to P, upon condition that he should pay it to B during the joint lives of A and B; it is evident that there would only remain to P, an annuity on the life of A after the decease of B: whence the truth of the proposition is manifest.

62. When any thing is affirmed or demonstrated of any life or lives, it is to be understood as applying equally to any proposed single life, or to the joint continuance of the whole of any number of lives that may be proposed together, or to that of any assigned number of the last survivors of them, or to the last surviving life of the whole.

63. Prop. 6. The present value of the reversion of a perpetual annuity after the failure of any life or lives, is equal to the excess of the present value of the perpetuity with immediate possession, above the present value of an annuity on the same life or lives.

64. Demons. If a perpetual annuity with immediate possession were granted to P, upon condition that he should pay the annual produce to another individual, during the existence of the life or lives proposed; it is evident that there would only remain to P, the reversion after the failure of such life or lives; and the present value of that reversion would manifestly be as stated above.

65. The 6th, 7th, and 8th tables at the end of this article, which have been extracted from the 19th, 21st, and 22d, respectively, in Mr Milne's Treatise on Annuities, will serve to illustrate the application of these propositions to the solution of questions in numbers.

In all the following examples, we suppose the lives to be such, as the general average of those the Carlisle table of mortality was constructed from, and the rate of interest to be 5 per cent.

66. Ex. 1. What is the present value of an annuity on the joint lives, and the life of the survivor of two persons now aged 40 and 50 years respectively?

According to No. 48, the process is as follows:

Value of a single life of \( \{ \begin{array}{c} 40 \\ 50 \end{array} \} \) by Table VI.

Subtract the value of their joint lives,

\[ \begin{align*} \text{sum} & = 25.050 \\ \text{remains} & = 15.066 \text{ years' purchase}, \end{align*} \]

the required value.

And if the annuity be L. 200, its present value will be L. 3013.2, or L. 3013.4s.

67. Ex. 2. The lives A, B, and C, being now aged 50, 55, and 60 years respectively, an annuity on the joint continuance of all the three, is worth 6.289 years purchase: What is the value of an annuity on the joint existence of the last two survivors of them?

According to No. 50, the process is thus:

| Ages | Values | |------|--------| | 50 & 55 | 8.528 | | 55 & 60 | 7.106 | | 50 & 60 | 7.601 |

Table VII.

Table VIII.

\[ \begin{align*} \text{sum} & = 23.235 \\ \text{Subtract} & = 2 \times 6.289 = 12.578 \\ \text{remains} & = 10.657 \text{ years' purchase, the required value}. \end{align*} \]

Therefore, if the annuity were L. 300, it would be worth L. 3197.2s. in present money.

68. Ex. 3. Required the value of an annuity on the last survivor of the three lives in the last example.

Proceeding according to No. 52, we have

| Ages | Values | |------|--------| | 50 | 11.660 | | 55 | 10.347 | | 60 | 8.940 | | 50, 55, & 60 | 6.289 |

Tab. VI.

(No. 67).

\[ \begin{align*} \text{sum} & = 37.236 \\ \text{Subtract the sum of the values of annuities on the three combinations of two lives,} & = 23.235 \\ \text{remains} & = 14.001 \text{ years' purchase, the required value}. \end{align*} \]

And if the annuity were L. 300, it would now be worth L. 4200.6s.

69. Ex. 4. What is the present value of an annuity on a life now 45 years of age, which is not to be entered upon until the expiration of ten years; the first payment thereof being to be made at the expiration of eleven years from this time, if the life survive till then?

Solution.

The present value of an annuity on a life of 55 is 10.347 (Table VI.), and the present value of L. 1 to be received upon the proposed life attaining to the age of 55, is \( \frac{4073}{4727} \times 0.613913 \); therefore, by No. 55, the required value is \( \frac{4073 \times 0.613913 \times 10.347}{4727} = 5.473 \) years purchase; so that if the annuity were L. 200, its present value would be L. 1094, 12s.

70. Ex. 5. Required the present value of an annuity to be received for the next ten years, provided that a person now 45 years of age, shall so long live.

Solution.

The present value of an annuity on a life of 45, to be entered upon immediately, is 12.648 (Table VI). Subtract that of an annuity on the same life deferred 10 years,

\[ \begin{align*} &\text{the remainder, } \\ &\quad 7.175 \\ \end{align*} \]

is the required number of years purchase. And, if the annuity were L. 200, its present value would be L. 1435.

71. Ex. 6. An annuity on a life of 45, deferred 10 years, was shown in No. 69, to be worth 5.473 years purchase in present money; let it be required to determine the equivalent annual payment for the same, to be made at the end of each of the next 10 years, but subject to failure upon the life failing in the term.

Solution.

The present value of L. 1 per annum on the proposed life for the next 10 years, has just been shown to be L. 7.175, and this, multiplied by the required annual payment, must produce L. 5.473; that payment must, therefore, be \( \frac{5.473}{7.175} = 0.76279 \). And, since the annual payment for the deferred annuity of L. 1 per annum would be L. 0.76279, that for an annuity of L. 200 must be L. 152, 11s. 2d.

72. Ex. 7. Let the present value be required of an annuity on a life now 40 years of age, to be payable only while that life survives another now of the age of 50 years.

From the present value of a life of 40,

\[ \begin{align*} &\text{Subtract that of the two joint lives, } \\ &\quad 9.984 \text{(Table VIII.)} \\ \end{align*} \]

the remainder,

\[ \begin{align*} &\text{years purchase is the required value (60).} \\ \end{align*} \]

Therefore, if the annuity were L. 100, it would be worth L. 340, 12s. in present money.

73. If the annuity in the last example were to be paid for by a constant annual premium at the end of each year while both the lives survived; by reasoning as in No. 71, it will be found, that such annual premium for an annuity of L. 1 should be

\[ \begin{align*} &\text{L. 0.341146; for an annuity of L. 100 it should therefore be L. 34, 2s. 3½d.} \\ \end{align*} \]

74. But if one of the equal premiums for this annuity is to be paid down now, and another at the end of each year while both the lives survive: the number of years purchase the whole of these premiums are worth, will evidently be increased by unity, on account of the payment made now, it will, therefore, be 10.984; and each premium for an annuity of L. 1 must, in this case, be

\[ \begin{align*} &\text{L. 0.3410087; for an annuity of L. 100 it should, therefore, be L. 31, 0s. 2d.} \\ \end{align*} \]

75. Ex. 8. Let it be required to determine the present value of the reversion of a perpetual annuity after the failure of a life now 50 years of age.

Solution.

The value of the perpetuity is 20 years purchase (8.)

\[ \begin{align*} &\text{Subtract that of an annuity on the life of 50, } \\ &\quad 11.660 \text{(Table VI.)} \\ \end{align*} \]

Remains 8.334 years purchase, the required value of the reversion (63.)

So that if the annuity were L. 300, its present value would be L. 2502.

76. In the same manner it will be found, by the 68th number and those referred to in the last example, that the reversion of a perpetuity, after the failure of the last survivor of three lives, now aged 50, 55, and 60 years respectively, is worth 5.999 years purchase in present money; therefore, if it were L. 100 per annum, its present value would be L. 599, 18s.

III. OF ASSURANCES ON LIVES.

77. An assurance upon a life, or lives, is a contract by which the Office or Underwriter, in consideration of a stipulated premium, engages to pay a certain sum upon such life or lives failing within the term for which the assurance is effected.

78. If the term of the assurance be the whole duration of the life or lives assured, the sum must necessarily be paid whenever the failure happens; and, in what follows, that payment is always supposed to be made at the end of the year in which the event assured against takes place. The anniversary of the assurance, or the day of the date of the policy, being accounted the beginning of each year.

79. At the end of the year in which any proposed life or lives may fail, the proprietor of the reversion of a perpetual annuity of L. 1 after their failure, will receive the pound then due, and will, at the same time, enter upon the perpetuity; therefore, the present value of the reversion is the same as that of L. 1 added to the money a perpetual annuity of L. 1 would cost, supposing this sum not to be receivable until the expiration of the year in which the failure of the life or lives might happen.

80. Hence we have this proportion. As the value of a perpetuity increased by unity is to L. 1, so is the present value of the reversion of a perpetual annuity of L. 1, after the failure of any life or lives, to the present value of L. 1, receivable at the end of the year in which such failure shall take place.

81. Therefore, if the value of an annuity of one pound on any life or lives, be subtracted from that of the perpetuity, and the remainder be divided by the value of the perpetuity increased by unity; the quo- tient will be the value, in present money, of the assurance of one pound on the same life or lives.

82. Ex. 1. What is the present value of L. 1, to be received at the end of the year, in which a life now 50 years of age may fail?

The rate of interest being 5 per cent., the value of the perpetuity is 20 years purchase, and that of the life 11·66; the answer therefore is \( \frac{20 - 11\cdot66}{20 + 1} = \frac{8\cdot34}{21} = L.0\cdot397143 \). And if the sum assured were L.1000, the present value of the assurance would be L.397, 2s. 10d.

83. When the term of a life assurance exceeds one year, its whole value is hardly ever paid down at the time that the contract is entered into, but, in the instrument (called a Policy) whereby the assurance is effected, an equivalent annual premium is stipulated for, payable at the commencement of each year during the term, but subject to failure with the life or lives assured.

84. But by reasoning as in No. 74, it will be found, that an annual premium payable at the commencement of each year in the whole duration of the life or lives assured, will be worth one year's purchase more, than an annuity on them payable at the end of each year; and, consequently, that if the value in present money of an assurance on any life or lives, be divided by the number of years purchase an annuity on the same life or lives is worth, increased by unity, the quotient will be the equivalent annual premium for the same assurance.

85. Ex. 2. Required the annual premium for the assurance of L.1, on a life of 50 years of age.

In No. 82, the single premium for that assurance was shown to be 0·397143, and the value of an annuity on the life is 11·66, therefore, by the preceding number, the required annual premium will be

\[ \frac{0\cdot397143}{12\cdot66} = 0\cdot313699 \text{ for the assurance of L.1;} \]

and for the assurance of L.1000, it will be L.31, 7s. 5d.

86. Ex. 3. Let both the single payment in present money, and the equivalent annual premium be required for the assurance of L.1, on the joint continuance of two lives of the respective ages of 45 and 50 years.

The value of an annuity of L.1 on the joint continuance of these two lives, appears by Table VII. to be L.9·737, therefore

\[ \frac{20 - 9\cdot737}{20 + 1} = \frac{10\cdot263}{21} = L.0\cdot488714 \text{ is the single premium, and } \frac{0\cdot488714}{10\cdot737} = L.0\cdot0455168, \text{ the equivalent annual one for the assurance of L.1 to be paid at the end of the year, in which that life becomes extinct which may happen to fail the first of the two.} \]

Therefore, if the sum assured were L.500, the total present value of the assurance would be L.244, 7s. 2d. and the equivalent annual premium L.22, 15s. 2d.

87. Ex. 4. Let both the single and the equivalent annual premium be required for the assurance of L.1, on the life of the survivor of two persons now aged 40 and 50 years respectively.

The value of an annuity on the survivor of these two lives was shown in No. 66, to be 15·066; therefore, by No. 81, the single premium will be

\[ \frac{20 - 15\cdot066}{20 + 1} = \frac{4\cdot934}{21} = L.0\cdot234952; \text{ and by No. 84, the annual one will be } \frac{L.0\cdot234952}{16\cdot066} = L.0\cdot0146942. \]

That is, for the assurance of L.1 to be received at the end of the year, in which the last surviving life of the two becomes extinct.

Therefore, for the assurance of L.5000, the single premium will be L.1174, 15s. 2d. the equivalent annual one L.73, 2s. 5d.

88. Ex. 5. What should the single and equivalent annual premiums be for an assurance on the last survivor of three lives, of the respective ages of 50, 55, and 60 years.

The value of an annuity on the last survivor of them, was shown in No. 68, to be 14·001, the single premium should therefore be

\[ \frac{20 - 14\cdot001}{20 + 1} = \frac{5\cdot999}{21} = L.0\cdot285666, \text{ and the annual } \frac{L.0\cdot285666}{15\cdot001} = L.0\cdot0190431, \text{ for the assurance of L.1, to be received at the end of the year, in which the last surviving life of the three may fail.} \]

For the assurance of L.2000, the single premium would therefore be L.571, 6s. 8d. the annual one L.38, 1s. 9d.

89. Lemma. If an annuity be payable at the commencement of each year, which some assigned life or lives may enter upon in a given term; the number of years purchase in its present value, will exceed by unity the number of years purchase, in the present value of an annuity on the same life or lives for one year less than the given term, but payable as annuities generally are, at the end of each year.

Demonstration. Since the proposed life or lives can only enter upon any year after the first, by surviving the year that precedes it; the receipt of each of the payments after the first, that are to be made at the commencement of the year, will take place at the same time, and upon the same conditions as that of the rent for the year then expired of the life-annuity, for a term one year less than the term proposed: this last mentioned annuity, will therefore, be worth in present money, just the same number of years' purchase as all the payments subsequent to the first, which may be made at the commencements of the several years.

And, since the first of these is to be made immediately, the present value of the whole of them, will evidently exceed the number of years purchase last mentioned, by unity, which was to be demonstrated.

90. If, while the rest remains the same, the payment of the annuity which depends upon the assigned life or lives entering upon any year, is not to be made until the end of that year; as the condition upon which every payment is to be made, will remain the same, but each of them will be one year later; the only alteration in the value of the whole, will arise from this increase in the remoteness of payment, by which it will be reduced in the ratio of L.1, to the present value of L.1, receivable at the end of a year (2). 91. When the value of an annuity on any proposed life or lives for an assigned term is given, it is evident that the value of an annuity on the same life or lives for one year less may be found, by subtracting from the given value, the present value, of the rent to be received upon the proposed life or lives surviving the term assigned.

92. Proposition. The present value of an assurance on any proposed life or lives for a given term, is equal to the excess of the value of an annuity to be paid at the end of each year, which the life or lives proposed may enter upon in the term, above the value of an annuity on them for the same term, but dependent, as usual, upon their surviving each year.

Demonstration. If an annuity payable at the end of each year, which the proposed life or lives may enter upon during the given term, be granted to P, upon condition that he shall pay over what he receives to Q, at the end of each year which the same life or lives may survive; it is manifest that there will only remain to P, the rent for the year in which the proposed life or lives may fail; that is, the assurance of that sum thereon for the given term (77). Which was to be demonstrated.

93. From the last four numbers (89—92) we derive the following

Rule for determining the present value of an assurance on any life or lives for a given term.

Add unity to the value of an annuity on the proposed life or lives, for the given term, and from the sum subtract the present value of one pound, to be received upon the same life or lives surviving the term; multiply the remainder by the present value of L.1, to be received certainly at the end of a year, and from the product subtract the present value of an annuity on the proposed life or lives for the term.

This last remainder will be the value in present money of the assurance of L.1 during the same term, on the life or lives proposed.

94. It has been shown above (34—39), how the present value of L.1, receivable upon any single or joint lives surviving an assigned term, may be found. And all that was demonstrated from No. 48. to 53. inclusive, respecting annuities on the last survivor of two, or of three lives, or on the joint continuance of the two last survivors out of three lives, is equally true of any particular year’s rent of those annuities. Hence it is evident, how the present value of L.1, to be received upon the last survivor of two, or of three lives, or upon the last two survivors out of three lives, surviving any assigned term, may be found.

95. Example. Required the present value of L.1, to be received at the end of the year, in which a life, now forty-five years of age, may fail, provided that such failure happens before the expiration of ten years.

Here the present value of L.1, to be received upon the life surviving the term, will be found to be L.0·528976, and the value of an annuity on the proposed life for the term, is 7·175 (70.)

From 8·175 subtract 0·528976 the remainder 7·646024 being multiplied by 0·952381 produces 7·28193 from this subtract 7·17500 remains L.0·10693, the required value of the assurance; and if the sum assured were L.3000, the value of the assurance in present money would be L.320, 15s. 7d.

96. By numbers 89, 91, and 95, it appears, that an annuity, payable at the commencement of each of the next ten years that a life of 45 enters upon, is worth 7·646 years purchase: therefore, \( \frac{0·10693}{7·646} = \)

L.0·013985 will be the annual premium for the assurance of L.1 for ten years on that life. For the assurance of L.3000, it will therefore be L.41, 19s. 1d.

97. When the term of the assurance is the whole duration of the life or lives assured, L.1 to be received upon their surviving the term is worth nothing; and an annuity on the lives for the term, is also for their whole duration.

Therefore, from No. 93. we derive the following

Rule for determining the present value of an assurance on any life or lives.

Add unity to the value of an annuity on the proposed life or lives; multiply the sum by the present value of L.1, to be received certainly at the end of a year; and from the product, subtract the value of an annuity on the same life or lives.

The remainder will be the value of the assurance in present money.

98. Example. Required the present value of L.1, to be received at the end of the year, in which the survivor of two lives may fail, their ages now being 40 and 50 years respectively.

The value of an annuity on these lives was shown in No. 66. to be 1·5066.

Multiply 1·5066 by 0·952381, from the product 1·43009, subtract 1·5066, the remainder L.0·2349 is the required value, agreeably to No. 87.

And, in all other cases, the values determined by the rule in the preceding number, will be found to agree with those obtained by the method of No. 81.

99. When an assurance on any life or lives has been effected at a constant annual premium, and kept up for some time, by the regular payment of that premium; the annual premium required for a new assurance of the same sum on the same life or lives, will, on account of the increase of age, be greater than that at which the first assurance was effected: Therefore, the present value of all these greater annual premiums, that is, the total present value of the new assurance, will exceed the present value of all the premiums that may hereafter be received under the existing policy. And the excess will evidently be the value of the policy, supposing the life or lives to be still insurable; that being the only advantage that can now be derived from the premiums already paid.

So that, if the present value of all the future annual premiums to be paid under an existing policy for the assurance of a certain sum upon any life or lives, be subtracted from the present value of the assurance of the same sum on the same life or lives; the remainder will be the value of the policy.

100. Example. L1000 has been assured some years, on a life now 50 years of age, for its whole duration, at the annual premium of L20, one of which has just now been paid: What is the value of the policy?

The present value of the assurance of L1000 on that life, has been shown in No. 82. to be L397, 2s. 10d.; and an annuity on the life, being worth 11'66 years purchase (Table VI.), the present value of all the premiums to be paid in future under the existing policy, is 11'66 × L20 = L233, 4s. 0d.; the value of the policy, therefore, is L163, 18s. 10d.

Immediately before the payment of the premium, the value of the policy would evidently have been less by the premium then due.

101. In our investigations of the values of annuities on lives, we have hitherto assumed, that no part of the rent is to be received for the year in which the life wherewith the annuity may terminate fails.

But if a part of the annuity is to be received at the end of that year, proportional to the part of the year which may have elapsed at the time of such failure; as, in a great number of such cases, some of the lives wherewith the annuity terminates will fail in every part of the year, and as many, or very nearly so, in any one part of it as in any other: we may assume, that, upon an average, half a year's rent will be received at the end of the year in which such failure happens; and, therefore, that by the title to what may be received after the failure of the life or lives whereon the annuity depends, the present value of that annuity will be increased by the present value of the assurance of half a year's rent on the same life or lives.

102. Thus, for example: the present value of the assurance of L1 on a life of 50 years of age, having in No. 82. been shown to be L0'397143, the value of an annuity of L1 on that life, when payable, till the last moment of its existence, will exceed L11'66, its value, if only payable, until the expiration of the last year it survives, by \( \frac{L_0'397143}{2} = L_0'199 \); it will therefore be L11'859.

103. If, at the end of the year in which an assigned life A may fail, Q or his heirs are to receive L1; and are, at the same time, to enter upon an annuity of L1, to be enjoyed during another life P, to be then fixed upon: the present value of Q's interest will evidently be the same as that of the assurance on the life of A, of a number of pounds, exceeding by unity the number of years purchase in the value of an annuity on the life of P, at the time of nomination.

104. What is the present value of the next presentation to a living of the clear annual value of L500; A, the present incumbent, being now 50 years of age; supposing the age of the clerk presented to be 25, at the end of the year in which the present incumbent dies; also, that he takes the whole produce of the living for that year?

By Table VI., it will be found, that the value of an annuity of L1 on a life of 25, is L15'303; and in No. 82. it has been shown, that the present value of the assurance of L1 on a life of 50, is L0'397143. Hence, and by the last number, it appears, that if the annual produce of the living were but L1, the present value of the next presentation would be L15'303 × L0'397143 = L6'47467. The required value, therefore, is L3237, 6s. 9d.

105. If, to the value of the succeeding life, determined according to No. 103, the value of the present be added, the sum of these will evidently be the present value of both the lives in succession; and, in the case of the preceding number, will be 6'575 + 11'66 = 18'135 years' purchase.

106. In No. 103, we proceeded upon the supposition that the annuity on the present life is only payable up to the expiration of the last year it survives; and, consequently, that the succeeding life takes the whole rent for the year in which the present fails.

But, if the succeeding life is only to take a part of that rent, in the same proportion to the whole, as the portion of the year which intervenes between the failure of the present life and the end of the year, is to the whole year; then, by reasoning as in No. 101, it will be found, that the portion of that rent which the succeeding life will receive, may be properly assumed to be one half. And, instead of increasing the number of years' purchase the annuity on the succeeding life will be worth at the end of the year in which the other fails, by unity, we must only add one half to that number, in order that the present value of the assurance of the sum on the existing life, may be the number of years' purchase, which all that may be received during the succeeding life, is now worth.

107. The value of the succeeding life, in the case of No. 104, will, upon this hypothesis, be 15'803 × 0'397143 = 6'27605 years' purchase.

And this appears to be the most correct way of calculating the value of an annuity on a succeeding life; although that of No. 103, proceeds upon the principle on which life interests are generally valued.

108. But the value of two lives in succession, will be the same on both hypotheses. The rent for the year in which the first may fail, being, in the one case, given entirely to the successor; and, in the other, divided equally between the two.

This is also true of any greater number of successive lives. PART II.

109. We now proceed to treat the subject of Annuities Algebraically.

I. ON ANNUITIES CERTAIN.

Let \( r \) denote the simple interest of L. 1 for one year.

\( s \), any sum put out at interest.

\( n \), the number of years for which it is lent.

\( m \), its amount in that time.

\( a \), an annuity for the same time (3 and 4.)

\( m \), the amount to which that annuity will increase, when each payment is laid up as it becomes due, and improved at compound interest until the end of the term.

\( v \), the present value of the same annuity (6.)

110. Then, reasoning as in the first number of this article, it will be found that \( m = s(1 + r)^n \). And by No. 2, it appears, that the present value of \( s \) pounds to be received certainly at the expiration of \( n \) years, is \( s \frac{1}{(1 + r)^n} \), or \( s(1 + r)^{-n} \).

111. The amount of L. 1 in \( n \) years being \( (1 + r)^n \), its increase in that time will be \( (1 + r)^n - 1 \), and when it is considered that this increase arises entirely from the simple interest (\( r \)) of L. 1 being laid up at the end of each year, and improved at compound interest during the remainder of the term; it must be obvious that \( (1 + r)^n - 1 \) is the amount of an annuity of \( r \) pounds in that time, but \( r : a :: (1 + r)^n - 1 : \frac{a}{r}[(1 + r)^n - 1] \), which, therefore, is equal to \( m \), the amount of an annuity of \( a \) pounds in \( n \) years.

112. Reasoning as in No. 8, it will be found, that since \( r : 1 :: a : \frac{a}{r} \), the present value of a perpetual annuity of \( a \) pounds is \( \frac{a}{r} \).

113. If two persons, \( A \) and \( B \), purchase a perpetuity of \( a \) pounds between them, which \( A \) and his heirs or assigns are to enjoy during the first \( n \) years, and \( B \) and his heirs and assigns, for ever after. Since the value of the perpetuity to be entered upon immediately, has just been shown to be \( \frac{a}{r} \), the present value of \( B \)'s share, that is, the present value of the same perpetuity when the entrance thereon is deferred until the expiration of \( n \) years, will be \( \frac{a}{r}(r + 1)^{-n} \), (110); and the value of the share of \( A \) will be thus much less than that of the whole perpetuity (21), and therefore equal to \( \frac{a}{r}[1 - (1 + r)^{-n}] = v \), the present value of an annuity of \( a \) pounds for the term of \( n \) years certain.

114. If the annuity is not to be entered upon until the expiration of \( d \) years, but is then to continue \( n \) years, its value at the time of entering upon it will be \( \frac{a}{r}[1 - (1 + r)^{-n}] \), as has just been shown; therefore its present value will be \( \frac{a}{r}[(1 + r)^{-d} - (1 + r)^{-(d + n)}] = v \), (110.)

115. In the same manner, it appears that, when the entrance on a perpetuity of \( a \) pounds is deferred \( d \) years, its present value will be \( \frac{a}{r}(1 + r)^{-d} \) (110 and 112.)

116. \( q \) being any number whatever, whole, fractional, or mixed, let \( \lambda q \) denote its logarithm, and \( xq \) the arithmetical complement of that logarithm; so that these equations may obtain, \( \frac{1}{q} = -\lambda q = xq \).

Then, for the resolution of the principal questions of this kind by logarithms, we shall have these formulae:

1. Amount of a sum improved at interest. \[ \lambda m = n\lambda(1 + r) + \lambda s, \quad (110.) \]

2. Amount of an annuity when each payment is laid up as it becomes due, and improved at interest until the expiration of the term. \[ \lambda m = \lambda[(1 + r)^n - 1] + \lambda a + xr, \quad (111.) \]

3. Value of a lease or an annuity. \[ \lambda v = \lambda[1 - (1 + r)^{-n}] + \lambda a + xr \quad (115.) \]

4. Value of a deferred annuity, or the renewal of any number of years lapsed in the term of a lease. \[ \lambda v = \lambda[(1 + r)^{-d} - (1 + r)^{-(d + n)}] + \lambda a + xr, \quad (114.) \]

5. Value of a deferred perpetuity, or the recession of an estate in fee simple, after an assigned term. \[ \lambda v = \lambda a + xr + dx(1 + r), \quad (115.) \]

By means of each of these equations, it is manifest that any one of the quantities involved in it may be found, when the rest are given.

117. If the interest be convertible into principal \( p \) times in the year, at \( r \) equal intervals, since the interest of L. 1 for one of these intervals will be \( \frac{r}{p} \) (109), and the number of conversions of interest into principal in \( n \) years \( pn \); to adapt the formula in No. 110. to this case, we have only to substitute \( \frac{r}{p} \) for \( r \), and \( pn \) for \( n \), in the equation \( m = s(1 + r)^n \) there given, whereby it will be transformed to this, \( m = s(1 + \frac{r}{p})^{pn} \).

118. According as \( s \) is equal to 1, 2, 4, or infinite; that is, according as the interest is convertible into principal yearly, half-yearly, quarterly, or continually, let \( m \) be equal to \( x, y, q, \) or \( c \); so shall \[ y = s(1+r)^n, \] \[ n = s\left(1+\frac{r}{2}\right)^{2n}, \] \[ q = s\left(1+\frac{r}{4}\right)^{4n}, \] and \( c = s \cdot N \)

\( N \) being the number whereof \( nr \) is the hyperbolic logarithm, and \( nr \times 0.43429448 \) its logarithm in Briggs' System, and the common tables.

119. From No. 117 and 110, it follows, that the present value of \( s \) pounds to be received at the end of \( n \) years, when the interest is convertible into principal at \( r \) equal intervals in each year, is \( s \left(1 + \frac{r}{v}\right)^{-vn} \).

120. When the present values and the amounts of annuities are desired, let the interest be convertible into principal at \( r \) equal intervals in the year, while the annuity is payable at \( \sigma \) intervals therein, the amount of each payment being \( \frac{a}{\pi} \).

121. Case I. \( \mu \) being any whole number not greater than \( r \), let \( \frac{1}{\sigma} = \frac{\mu}{r} \), so that the interest may be convertible into principal \( \mu \) times in each of the intervals between the payments of the annuity.

Then will the amount of L.1, at the expiration of the period \( \frac{1}{\sigma} \) be \( \left(1 + \frac{r}{v}\right)^{\mu} \) (117), and the interest of L.1 for the same time will be \( \left(1 + \frac{r}{v}\right)^{\mu} - 1 \); whence the present value of the perpetuity will be

\[ \frac{1}{\pi} \frac{a}{\left(1 + \frac{r}{v}\right)^{\mu} - 1} \]

(8), and the value of the same deferred \( n \) years, will be \( \frac{a}{\pi} \frac{\left(1 + \frac{r}{v}\right)^{-vn}}{\left(1 + \frac{r}{v}\right)^{\mu} - 1} \) (119), therefore the present value of the annuity to be entered upon immediately, and to continue \( n \) years, will be

\[ \frac{a}{\pi} \frac{1 - \left(1 + \frac{r}{v}\right)^{-vn}}{\left(1 + \frac{r}{v}\right)^{\mu} - 1} = v. \]

122. Case 2. \( \mu \) being any whole number greater than \( r \), let \( \frac{1}{\sigma} = \frac{\mu}{r} \), so that the annuity may be payable \( \mu \) times in each of the intervals between the payments of interest, or the conversion thereof into principal.

Then, at the expiration of the \( \frac{1}{\sigma} \)-th of a year, when the interest on the purchase-money is first payable or convertible, the interest on all the \( \mu-1 \) payments of the annuity previously made, will be

\[ \frac{ar}{\pi} \left[ (\mu-1) + (\mu-2) + (\mu-3) + \cdots + 3 + 2 + 1 \right] = \frac{a}{\pi} \frac{r \mu (\mu-1)}{2\pi}; \text{to which, adding the } \mu \text{ payments of } \frac{a}{\pi} \text{ each (including the one only then due), the sum, } \frac{a}{\pi} \left[ \mu + \frac{r \mu (\mu-1)}{2\pi} \right], \text{ is the simple interest which the value of the perpetuity should yield at the expiration of each } r \text{th part of a year, in order to supply the deficiency (both of principal and interest) that would be occasioned during each of those periods, in any fund out of which the several payments of the annuity might be taken, as they respectively became due; and since } \frac{r}{v} : \frac{a}{\pi} \left[ \mu + \frac{r \mu (\mu-1)}{2\pi} \right] :: 1 : \frac{av}{\pi} \left[ \mu + \frac{r \mu (\mu-1)}{2\pi} \right] = a \left(1 + \frac{\mu-1}{2\pi}\right), \text{ this last expression will be the value of such perpetuity with immediate possession (8); the value of the same deferred } n \text{ years, will therefore be } a \left(1 + \frac{\mu-1}{2\pi}\right) \times \left(1 + \frac{r}{v}\right)^{-vn} \text{ (119). Whence it appears, that the present value of the annuity to be entered upon immediately, and to continue } n \text{ years, will be } a \left(1 + \frac{\mu-1}{2\pi}\right) \cdot \left[1 - \left(1 + \frac{r}{v}\right)^{-vn}\right] = v. \]

123. Case 3. When, in consequence of the annuity being always payable at the same time that the interest is convertible, \( r = \pi \).

Since the interest of L.1 at the expiration of the period \( \frac{1}{\sigma} \) will be \( \frac{r}{\pi} \), the value of the perpetuity will be

\[ \frac{1}{\pi} \frac{a}{\frac{r}{\pi}} = \frac{a}{r} \text{(8)}, \text{whence, proceeding as before, we obtain the present value of the annuity, } \frac{a}{r} \left[1 - \left(1 + \frac{r}{v}\right)^{-vn}\right] = v. \text{ When } r = \pi, \text{ and consequently } \mu = 1, \text{ the values of } v, \text{ given in the two preceding cases, will be found to coincide with this.}

124. According as \( \sigma \) and \( \pi \) are each equal to 1, 2, 4, or are infinite; that is, according as the interest and the annuity are each payable yearly, half-yearly, quarterly, or continually, let \( v \) be equal to \( y, h, q, \) or \( c \), then will

\[ y = \frac{a}{r} \left[1 - \left(1 + r\right)^{-n}\right], \] \[ h = \frac{a}{r} \left[1 - \left(1 + \frac{r}{2}\right)^{-2n}\right], \] \[ q = \frac{a}{r} \left[1 - \left(1 + \frac{r}{4}\right)^{-4n}\right], \] and \( c = \frac{a}{r} \left[1 - \frac{1}{N}\right], \text{ } N \text{ being as in No. 118.}

125. The amount of an annuity is equal to the sum to which the purchase money would amount, if it were put out and improved at interest during the whole term.

For, from the time of the purchase of the annuity, whatever part of the money that was paid for it may be in the hands of the grantor, he must improve thus to provide for the payments thereof; and if the purchaser also improve in the same manner all he receives, the original purchase money must evident- Algebraically receive the same improvement during the term, as if it had been laid up at interest at its commencement.

126. The periods of conversion of interest into principal, and of the payment of the annuity being still designated as in No. 120; since in \( n \) years, the number of periods of conversion will be \( mn \), in the

1st Case, Where the interest is convertible \( \mu \) times in each of the intervals between the payments of the annuity, we have

\[ (1 + r)^n v = \frac{a}{r} \left( \frac{(1 + r)^n - 1}{r} \right) \]

\( m_1 (117, 121, \text{and } 125) \). In the 2d Case, when the annuity is payable \( \mu \) times, in each interval between the conversions of interest,

\[ (1 + r)^n v = \frac{a}{r} \left[ \frac{1}{r} \left( \frac{(1 + r)^n - 1}{r} \right) \right] \]

\( m_2 (117, 122, \text{and } 125) \).

And, in the 3d Case, when the annuity is always payable at the same time that the interest is convertible,

\[ (1 + r)^n v = \frac{a}{r} \left[ \frac{1}{r} \left( \frac{(1 + r)^n - 1}{r} \right) \right] = m_3 (117, 123, \text{and } 125). \]

127. According as \( n \) and \( r \) are each equal to 1, 2, 4, or are infinite; that is, according as the interest and the annuity are each payable yearly, half-yearly, quarterly, or continually, let \( m \) be denoted by \( y', h', q', \text{or } c' \);

then will

\[ y' = \frac{a}{r} \left[ \frac{(1 + r)^n - 1}{r} \right], \]

\[ h' = \frac{a}{r} \left[ \frac{(1 + r)^n - 1}{r} \right], \]

\[ q' = \frac{a}{r} \left[ \frac{(1 + r)^n - 1}{r} \right], \]

and \( c' = \frac{a}{r} (n-1); n \) being as in No. 118.

128. Example 1. What will L.320 amount to, when improved at compound interest during 40 years; the rate of interest being 4 per cent. per annum?

By the first formula in No. 116, the operation will be as follows:

\[ 1 + r = 1.04 \times 0.01703334 \]

\[ \times n = 40 \]

\[ (1 + r)^n = 0.6813336 \]

\[ s = 320 \times 2.5051500 \]

\( m = 1536.327 \times 3.1864836 \)

And the answer is L.1536, 6s. 6½d.

129. Ex. 2. If the interest were convertible into principal every half-year, the operation, according to No. 117, would be thus:

\[ 1 + r = 1.02 \times 0.00860017 \]

\[ \times n = 80 \]

\[ 0.6880136 \]

\[ s = 320 \times 2.5051500 \]

\( m = 1560.14 \times 3.1931636 \)

So that in this case the amount would be L.1560, 2s. 9½d.

130. Ex. 3. Required the present value of an annuity of L.250 for 30 years, reckoning interest at 5 per cent.

By the third formula in No. 116, the operation will be thus:

\[ \lambda (1 + r)^{-1} = x 1.05 = 1.9788107 \]

\[ \times n = 30 \]

\[ (1 + r)^{-n} = 23137704 \times 1.3643210 \]

\[ 1-(1+r)^{-n} = 76862296 \times 1.8857133 \]

\[ a = 250 \times 2.3979400 \]

\[ r = .05 \times 1.3010300 \]

\[ v = 3843.114 \times 3.5846833 \]

And the required value is L.3843, 2s. 3½d.

131. Ex. 4. The rest being still the same, if the annuity in the last example be payable half-yearly, in the formula of No. 122, \( r \) will be equal to 1, \( s = 2 \), and \( \mu = 2 \); that formula will therefore become \( a \left( \frac{1}{r} + \frac{1}{2} \right) \left[ 1-(1+r)^{-n} \right] = v \); and the operation will be thus:

\[ 1-(1+r)^{-n} = 1.8857133 \]

\[ a = 250 \times 2.3979400 \]

\[ \frac{1}{r} + \frac{1}{2} = 20.25 \times 1.3064250 \]

\[ v = 3891.15 \times 3.5900783 \]

The value of the annuity will, therefore, in this case, be L.3891, 3s.

132. Ex. 5. To what sum will an annuity of L.120 for 20 years amount, when each payment is improved at compound interest, from the time of its becoming due until the expiration of the term; the rate of interest being 6 per cent.?

The operation by the second formula in No. 116 is thus:

\[ 1 + r = 1.06 \times 0.025305865 \]

\[ \times n = 20 \]

\[ (1 + r)^n = 3.207135 \times 0.5061173 \]

\[ (1 + r)^n - 1 = 2.207135 \times 0.3438289 \]

\[ a = 120 \times 2.0791812 \]

\[ r = .06 \times 1.2218487 \]

\( m = 4414.27 \times 3.6448588 \)

And the amount required is L.4414, 5s. 6d.

133. Ex. 6. The rest being the same as in the last example; if both the interest and the annuity be payable half-yearly, the amount will be determined by the second of the formulæ given in No. 127; which, in this case, will become \( \frac{120}{0.6} \left[ (1.03)^{40} - 1 \right] \), and the operation will be as follows: II. ON THE PROBABILITIES OF LIFE.

134. Any persons \( A, B, C, \) &c. being proposed, let the numbers which tables of mortality (§2) adapted to them, represent to attain to their respective ages, be denoted by the symbols \( a, b, c, \) &c.; while lives \( n \) years older than those respectively, are denoted thus:

\[ \begin{align*} nA, &\quad nB, &\quad nC, &\quad \text{&c.} \\ \end{align*} \]

and the numbers that attain to their ages, by the symbols \( na, nb, nc, \) &c.; also let lives \( n \) years younger than \( A, B, C, \) &c. be denoted thus:

\[ \begin{align*} A_n, &\quad B_n, &\quad C_n, &\quad \text{&c.,} \\ \end{align*} \]

while the numbers which the tables show to attain to those younger ages, are designated by the symbols \( a_n, b_n, c_n, \) &c.

Then, if \( A \) be 21 years of age, and we use the Carlisle Table, we shall have \( a = 6047, \) and \( na = 5362, \) the number that attain to the age of thirty-five, or that live to be fourteen years older than \( A. \)

Hence the number that are represented by the table to die in \( n \) years from the age of \( A, \) will be

\[ \begin{align*} a - na, &\quad \text{that is in } 14 \text{ years, } a - \frac{1}{14}a; \\ \end{align*} \]

and by the Carlisle Table, \( n \) 14 years from the age of 21, that is, between 21 and 35; it will be \( 6047 - 5362 = 685. \)

135. Problem. To determine the probability that a proposed life \( A, \) will survive \( n \) years.

Solution. \( a \) being the number of lives in the table of mortality, that attain to the age of that which is proposed, conceive that number of lives to be so selected, (of which \( A \) must be one,) that they may each have the same prospect with regard to longevity, as the proposed life and those in the table, or the average of those from which it was constructed; then will the number of them that survive the term be \( na \) (134).

So that the number of ways all equally probable, or the number of equal chances for the happening of the event in question is \( na; \) and the whole number for its either happening or failing is \( a; \) therefore, according to the first principles of the doctrine of probabilities, the probability of the event happening, that is, of \( A \) surviving the term, is \( \frac{na}{a}. \)

If the age of \( A \) be 14, the probability of that life surviving 7 years, or the age of 21, will, according to the Carlisle Table, be \( \frac{na}{a} = \frac{6047}{6335} = 0.95454. \)

136. Since the number that die in \( n \) years from the age of \( A \) is \( a - na \) (134), it appears, in the same manner, that the probability of that life failing in \( n \) years will be \( \frac{a - na}{a} = 1 - \frac{na}{a}, \) which probability, when the life, term, and table of mortality, are the same as in the last No. will be 0.04546.

137. If two lives \( A \) and \( B \) be proposed, since the probability of \( A \) surviving \( n \) years will be \( \frac{na}{a}, \) and that of \( B \) surviving the same term will be \( \frac{nb}{b}; \) it appears from the doctrine of probabilities that \( \frac{na}{a} \cdot \frac{nb}{b} \) or \( \frac{n(ab)}{ab} \) will be the measure of the probability that these lives will both survive \( n \) years.

In the same manner it may be shown, that the probability of the three lives \( A, B, \) and \( C \) all surviving \( n \) years, will be measured by \( \frac{na}{a} \cdot \frac{nb}{b} \cdot \frac{nc}{c}, \) or \( \frac{n(abc)}{abc}. \) And, universally, that any number of lives \( A, B, C, \) &c. will jointly survive \( n \) years, the probability is \( \frac{n(abc, \text{&c.})}{abc, \text{&c.}} \)

138. Let \( \frac{na}{a} = a, \frac{nb}{b} = b, \frac{nc}{c} = c, \) &c.; also let \( \frac{n(abc, \text{&c.})}{abc, \text{&c.}} = (abc, \text{&c.}); \) so that the probabilities of \( A, B, C, \) &c. surviving \( n \) years may be denoted by \( na, nb, nc, \) &c. respectively; and that of all those lives jointly surviving that term by \( n(abc, \text{&c.}) \)

Then will the probability that none of those lives will survive \( n \) years, be \( (1 - a) \cdot (1 - b) \cdot (1 - c), \) &c.

139. But the probability that some one or more of these lives will survive \( n \) years, will be just what the probability last mentioned is deficient of certainty; its measure therefore, being just what the measure of that probability is deficient of unity, will be

\[ 1 - (1 - a) \cdot (1 - b) \cdot (1 - c), \text{ &c.} \]

140. Corol. 1. When there is only one life \( A, \) this will be \( na. \)

141. Corol. 2. When there are two lives \( A \) and \( B, \) it becomes \( na + nb - nab. \)

142. Corol. 3. When there are three lives \( A, B, \) and \( C, \) it becomes \( na + nb + nc - nab - nc - (bc) + (abc). \)

143. When three lives \( A, B, \) and \( C \) are proposed, that at the expiration of \( n \) years there will be

Algebraical living | dead | the probability is ---|---|--- ABC | none | \( \frac{(abc)}{abc} \)

\( AB \) | \( C \) | \( (ab) \cdot (1 - c) = (ab) - (abc) \) \( AC \) | \( B \) | \( (ac) \cdot (1 - b) = (ac) - (abc) \) \( BC \) | \( A \) | \( (bc) \cdot (1 - a) = (bc) - (abc) \)

And the sum of these four \( n(ab) + n(ac) + n(bc) - 2(abc) \), is the measure of the probability that some two at the least, out of these three lives, will survive the term.

III. OF ANNUITIES ON LIVES.

144. Let the number of years purchase that an annuity on the life of \( A \) is worth, that is, the present value of L. 1, to be received at the end of every year during the continuance of that life, be denoted by \( A \); while the present value of an annuity on any number of joint lives \( A, B, C, \) &c., that is, of an annuity which is to continue during the joint existence of all the lives, but to cease with the first that fails, is denoted by \( ABC, \) &c.

Then will the value of an annuity on the joint continuance of the three lives \( A, B, \) and \( C, \) be denoted by \( ABC. \)

And on the joint continuance of the two \( A \) and \( B, \) by \( AB. \)

145. Also let \( tA \) and \( A_t \) denote the value of annuities on lives respectively older and younger than \( A, \) by \( t \) years; While \( (ABC, \) &c.) designates the value of an annuity on the joint continuance of lives \( t \) years older than \( A, B, C, \) &c. respectively; and \( (ABC, \) &c.)\( t \) that of an annuity on the same number of joint lives, as many years younger than these respectively.

146. Let \( \frac{1}{1+r} \), the present value of L. 1 to be received certainly at the expiration of a year, be denoted by \( v. \)

Then will \( v^n \) be the present value of that sum certain to be received at the expiration of \( n \) years.

But if its receipt at the end of that time, be dependent upon an assigned life \( A, \) surviving the term, its present value will, by that condition, be reduced in the ratio of certainty to the probability of \( A \) surviving the term, that is, in the ratio of unity to \( n^a, \) and will therefore be \( \frac{av^n}{n^a}. \)

In the same manner it appears, that if the receipt of the money at the expiration of the term be dependent upon any assigned lives, as \( A, B, C, \) &c. jointly surviving that period, its present value will be \( \frac{(abc &c.)v^n}{n^a}. \)

147. Let us denote the sum of any series, as \( 1(abc)v + 2(abc)v^2 + 3(abc)v^3 + \) &c. thus, \( S_n(abc)v^n, \) by prefixing the italic capital \( S \) to the general term thereof. Then, from what has just been advanced, it will be evident, that \( ABC, \) &c. \( = \frac{S_n(abc &c.)v^n}{abc &c.}. \)

When there are but three lives \( A, B, \) and \( C; \) this becomes \( ABC = S_n(abc)v^n. \)

When there are but two, \( A \) and \( B, \) it becomes \( AB = S_n(ab)v^n. \)

And in the same manner it appears, that for a single life \( A, \) \( A = S_nav^n. \)

148. \( \frac{n(abc &c.)v^n}{abc &c.} = \frac{n(abc &c.)v^n}{abc &c.} \) (138), where the denominator \( (abc &c.) \) is constant, while the numerator varies with the variable exponent \( n. \) And the most obvious method of finding the value of an annuity on any assigned single or joint lives, is to calculate the numerical value of the term \( \frac{n(abc &c.)v^n}{abc &c.} \) for each value of \( n, \) and then to divide the sum of all these values by \( abc &c.; \) for \( \frac{S_n(abc &c.)v^n}{abc &c.} = ABC &c. \)

In calculating a table of the values of annuities on lives in that manner, for every combination of joint lives, it would be necessary to calculate the term \( \frac{n(abc &c.)v^n}{abc &c.} \) for as many years as there might be between the age of the oldest life involved and the oldest in the table; and the same number of the terms \( nav^n \) for any single life of the same age.

But this labour may be greatly abridged as follows:

PROB. I.

149. Given \( (ABC, \) &c.), the value of an annuity on any number of joint lives, to determine \( ABC, \) &c. that of an annuity on the same number of joint lives respectively one year younger than them.

Solution.

If it were certain that the lives \( ABC, \) &c. would all survive one year, the proprietor of an annuity of L.1, dependent upon their joint continuance, would, at the expiration of a year, be in possession of L.1, (the first year's rent,) and an annuity on the same number of lives, one year older respectively than \( ABC, \) &c. therefore, in that case, the required present value of the annuity would be \( v[1 + (ABC, \) &c.)], (146.)

But the probability of the lives \( A, B, C, \) &c. jointly surviving one year, is less than certainty, in the ratio of \( (abc &c.) \) to unity; therefore \( ABC &c. = (abc &c.)v[1 + (ABC &c.)]. \)

150. Corol. 1. When there are but three lives, \( A, B, \) and \( C, \) this becomes \( ABC = (abc)v[1 + (ABC)]. \)

151. Corol. 2. When there are only two, \( A \) and \( B, \) \( AB = (ab)v[1 + (AB)]. \)

152. Corol. 3. And for a single life \( A, \) it appears, in the same manner, that \( A = av(1 + 'A). \)

153. Hence, in logarithms, we have these equations,

Upon which it may be observed, that \( \lambda v + \lambda a \), the sum of the first two logarithms that are employed in determining \( \lambda \) from \( \lambda a \), also enters the operation whereby \( \lambda b \) is determined from \( \lambda (ab) \). And that \( \lambda v + \lambda a + \lambda b \), the sum of the first three logarithms that serve to determine \( \lambda b \) from \( \lambda (ab) \), is also required to determine \( \lambda c \) from \( \lambda (abc) \); which observation may be extended in a similar manner to any greater number of joint lives.

154. By these means it is easy to complete a table of the values of annuities on single lives of all ages; beginning with the oldest in the table, and proceeding regularly age by age to the youngest.

Also a table of the values of any number of joint lives, the lives in each succeeding combination, in any one series of operations, (according to the retrograde order of the ages in which they are computed), being one year younger respectively than those in the preceding combination.

And, if a table of single lives be computed first, then of two joint lives, next of three joint lives, and so on; the calculations made for the preceding tables will be of great use for the succeeding.

155. Having shown how to compute tables of the values of annuities on single and joint lives, we shall, in what follows, always suppose those values to be given.

156. Let the value of an annuity on the joint continuance of any number of lives, \( A, B, C, \ldots \) that is not to be entered upon until the expiration of \( t \) years be denoted by \( t(ABC \&c.) \).

Then, if it were certain that all the lives would survive the term, since the value of the annuity at the expiration of the term would be \( t(ABC \&c.) \), its present value would be \( v^t \cdot t(ABC \&c.) \).

But the measure of the probability that all the lives will survive the term is \( t(abc \&c.) \), therefore

\[ t(ABC \&c.) = t(abc \&c.) \cdot v^t \cdot t(ABC \&c.). \]

In the same manner, it appears, that for a single life \( A \),

\[ t(A) = av^t \cdot t(A). \]

157. Let an annuity for the term of \( t \) years only, dependent upon the joint continuance of any number of lives, \( A, B, C, \ldots \) be denoted by \( t(ABC \&c.) \); and, since this temporary annuity, together with an annuity on the joint continuance of the same lives deferred for the same term, will evidently be of the same value as an annuity to be entered upon immediately, and enjoyed during their whole joint continuance, we have \( t(ABC \&c.) + t(ABC \&c.) = ABC \&c.; \) whence,

\[ t(ABC \&c.) = ABC \&c.; \] and for a single life \( A \),

\[ t(A) = A - t(A). \]

PROB. II.

158. To determine the present value of an annuity on the survivor of the two lives \( A \) and \( B \), (155), which we designate thus, \( AB \).

Solution.

The probability that the survivor of these two lives will outlive the term of \( n \) years, was shown in No. 141, to be \( a + b - ab \); therefore, reasoning as in No. 146, it will be found, that the present value of the \( n \)th year's rent of this annuity is

\[ \left[ a + b - ab \right]^n v^n, \] and the value of all the rents thereof will be \( S \left[ a + b - ab \right]^n v^n \), or \( S_{av}^n + S_{bv}^n - S_{abv}^n \); so that \( AB = A + B - AB \) (147), agreeably to No. 48.

PROB. III.

159. To determine the present value of an annuity on the last survivor of three lives, \( A, B, \) and \( C \), (155); which we denote thus, \( ABC \).

Solution.

The present value of the \( n \)th year's rent is

\[ \left[ a + b + c - ab - ac - bc + abc \right]^n \] (142 and 146); whence, it appears, as in the preceding number, that \( ABC = A + B + C - AB - AC - BC + ABC \), agreeably to No. 52.

PROB. IV.

160. To determine the present value of an annuity on the joint existence of the last two survivors out of three lives, \( A, B, C \), (155); which we denote thus, \( ABC \).

Solution.

The present value of the \( n \)th year's rent is

\[ \left[ ab + ac + bc - 2abc \right]^n \] (143 and 146); whence, reasoning as in the two preceding numbers, we infer, that \( ABC = AB + AC + BC - 2ABC \), as was demonstrated otherwise in No. 51.

161. Since the solutions of the last three problems were all obtained by showing each year's rent (as for instance the \( n \)th) of the annuity in question, to be of the same value with the aggregate of the rents for the same year, of all the annuities (taken with their proper signs) on the single and joint lives exhibited in the resulting formula: if any term of years be assigned, it is manifest that the value of such annuity for the term, must be the same as that of the aggregate of the annuities above mentioned, each for the same term.

PROB. V.

162. \( A \) and \( B \) being any two proposed lives now both existing, to determine the present value of an annuity receivable only while \( A \) survives \( B \). A rent of this annuity will only be payable at the end of the \( n \)th year, provided that \( B \) be then dead, and \( A \) living; but the probability of \( B \) being then dead is \( \frac{1}{n} \), and that of \( A \) being then living \( a \), and these two events are independent; therefore, the probability of their both happening, or that of the rent being received, is \( (1 - \frac{b}{n})a = \frac{a}{n}(ab) \); the present value of that rent is, therefore, \( \left[ \frac{a}{n}(ab) \right]e^n \); whence, it follows, that the required value of the annuity on the life of \( A \) after that of \( B \), is \( A - AB \), agreeably to No. 60.

163. If the payment for the annuity which was the subject of the last problem, is not to be made in present money, but by a constant annual premium \( p \) at the end of each year, while both the lives survive; since \( AB \) is the number of years purchase (6) that an annuity on the joint continuance of those lives is worth, the value of \( p \) will be determined by this equation, \( p \cdot AB = A - AB \), whence we have \( p = \frac{A}{AB} - 1 \).

164. But if one premium \( p \) is to be paid down now, and an equal premium at the end of each year while both the lives survive, we shall have \( p \cdot (1 + AB) = A - AB \), and \( p = \frac{A - AB}{1 + AB} \).

165. For numerical examples illustrative of the formulae given from No. 158 to the present; see Nos. 66—74.

**Prob. vi.**

166. \( A \) and \( B \) are in possession of an annuity on the life of the survivor of them, which, if either of them die before a third person \( C \), is then to be divided equally between \( C \) and the survivor during their joint lives; to determine the value of \( C \)'s interest.

Solution.

That at the end of the \( n \)th year there will be

| dead | living | |------|--------| | \( A \) | \( BC \) | | \( B \) | \( AC \) |

The probability, multiplied by \( C \)'s proportion of the annuity in that circumstance, is

\[ \frac{1}{2} \left( \frac{(1-a)}{n}(bc) + \frac{1}{2} \left[ \frac{(bc)}{n}(abc) \right] \right) \]

and the sum of these being \( \frac{1}{2}(ac) + \frac{1}{2}(bc) - \frac{1}{2}(abc) \), the value of \( C \)'s interest is \( \frac{1}{2}AC + \frac{1}{2}BC - ABC \).

**Prob. vii.**

167. An annuity after the decease of \( A \), is to be equally divided between \( B \) and \( C \) during their joint lives, and is then to go entirely to the last survivor for his life; it is proposed to find the value of \( B \)'s interest therein.

That at the end of the \( n \)th year there will be

| dead | living | |------|--------| | \( A \) | \( BC \) | | \( AC \) | \( B \) |

The probability, multiplied by \( B \)'s proportion of the annuity in that circumstance, is

\[ \frac{1}{2} \left( \frac{(1-b)}{n}(ac) + \frac{1}{2} \left[ \frac{(ac)}{n}(abc) \right] \right) \]

and the sum of these being \( \frac{1}{2}(ac) + \frac{1}{2}(bc) - \frac{1}{2}(abc) \), the value of \( B \)'s interest is \( \frac{1}{2}AC + \frac{1}{2}BC - ABC \).

**Prob. viii.**

168. \( A \), \( B \), and \( C \) purchase an annuity on the life of the last survivor of them, which is to be divided equally at the end of every year among such of them as may then be living; what should \( A \) contribute towards the purchase of this annuity?

Solution.

That at the end of \( n \) years there will be

| dead | living | |------|--------| | none | \( ABC \) | | \( C \) | \( AB \) | | \( B \) | \( AC \) | | \( BC \) | \( A \) |

The probability, multiplied by \( A \)'s proportion of the annuity in that circumstance, is

\[ \frac{1}{2} \left( \frac{(1-a)}{n}(bc) + \frac{1}{2} \left[ \frac{(bc)}{n}(abc) \right] \right) \]

and the sum of these being \( \frac{1}{2}(ab) + \frac{1}{2}(ac) + \frac{1}{2}(abc) \), the required value of \( A \)'s interest is \( A - \frac{1}{2}AB - \frac{1}{2}AC + \frac{1}{2}ABC \).

**Prob. ix.**

169. As soon as any two of the three lives, \( A \), \( B \), and \( C \), are extinct, \( D \) or his heirs are to enter upon an annuity; which they are to enjoy during the remainder of the survivor's life; to determine the value of \( D \)'s interest therein.

Solution.

That at the end of \( n \) years there will be

| dead | living | |------|--------| | \( AB \) | \( C \) | | \( AC \) | \( B \) | | \( BC \) | \( A \) |

The probability is

\[ \frac{1}{2} \left( \frac{(1-a)}{n}(bc) + \frac{1}{2} \left[ \frac{(bc)}{n}(abc) \right] \right) \]

and the sum of all these being \( \frac{1}{2}(ab) + \frac{1}{2}(ac) + \frac{1}{2}(abc) \), the value of \( D \)'s interest is

\[ A + B + C - 2AB - 2AC - 2BC + 3ABC. \]

170. The last four may be sufficient to show the method of proceeding with any similar problems.

171. Let \( \frac{m}{(abc, \&c.)} \) denote the probability that the Algebraical last \( m \) survivors out of \( m + \mu \) lives \( A, B, C, \ldots \) will jointly survive the term of \( t \) years. And when \( \mu = 0 \), the expression will become \( (abc, \ldots) \) the probability that the lives will all survive the term (138).

When \( m = 1 \) it will become \( \frac{1}{(abc, \ldots)} \) the measure of the probability that the last survivor of them will outlive the term; which it will be better to write thus, \( (abc, \ldots) \), retaining the vinculum, but omitting the unit over it, as in the notation of powers.

Also let \( ABC, \ldots \) denote the value of an annuity on the joint continuance of the same number of last survivors out of the same lives. Then, if \( \mu \) be equal to 0, it will be \( ABC, \ldots \) the value of an annuity on the joint continuance of all the lives; when \( m = 1 \), it will be \( ABC, \ldots \) the value of an annuity on the last survivor of them. The values of annuities on the last survivor of two and of three lives, will be denoted as in Nos. 158 and 159 respectively; and that of an annuity on the joint continuance of the last two survivors out of three lives, as in No. 160.

The value of an annuity on the last \( m \) survivors out of these \( m + \mu \) lives, according as it is limited to the term of \( t \) years, or deferred during that term, will also be denoted by \( \frac{m}{ABC, \ldots} \) or \( \frac{m}{ABC, \ldots} \) (156 and 157.)

**Prob. X.**

172. An annuity certain for the term of \( t + r \) years, is to be enjoyed by \( P \) and his heirs during the joint existence of the last \( m \) survivors out of \( m + \mu \) lives, \( A, B, C, \ldots \); and if that joint existence fail before the expiration of \( t \) years, the annuity is to go to \( Q \) and his heirs, for the remainder of the term; to determine the value of \( Q \)'s interest in that annuity.

**Solution.**

\( Q \)'s expectation may be distinguished into two parts:

1st, That of enjoying the annuity during the term of \( t \) years. 2d, That of enjoying it after the expiration of that term.

The sum of the present values of the interests of \( P \) and \( Q \), together in the annuity for the term of \( t \) years, is manifestly equal to the whole present value of the annuity certain for that term; that is, equal to \( \frac{1-v^t}{r} \) (113 and 146); and the value of \( P \)'s interest for the term of \( t \) years, is \( \frac{m}{ABC, \ldots} \) (171); therefore the value of \( Q \)'s interest for the same term is \( \frac{1-v^t}{r} - \frac{m}{ABC, \ldots} \).

The present value of the annuity certain for \( r \) years after \( t \) years is \( \frac{v^t(1-v^r)}{r} \) (114 and 146); and \( Q \) and his heirs will receive this annuity, if the joint continuance of the last \( m \) survivors above mentioned fail before the expiration of \( t \) years; but the probability of their joint continuance failing in the term, is \( \frac{1}{(abc, \ldots)} \); therefore, the value of \( Q \)'s interest in the annuity to be received after \( t \) years, is

\[ \left[1-\frac{1}{(abc, \ldots)}\right] \frac{v^t(1-v^r)}{r}; \]

and the whole value of \( Q \)'s interest, is \( \frac{1}{r} \left[1-v^t - v^r(1-v^r)\right] \frac{m}{ABC, \ldots} \)

\[ = \frac{m}{ABC, \ldots} \]

173. Corol. 1. When the whole annuity certain is a perpetuity, \( v^t = 0 \), and the value of \( Q \)'s interest is \( \frac{1}{r} \left[1-\frac{1}{(abc, \ldots)}\right] \frac{m}{ABC, \ldots} \).

174. Corol. 2. When the term \( t \) is not less than the greatest joint continuance of any \( m \) of the proposed lives, according to the tables of mortality adapted to them, \( (abc, \ldots) = 0 \), and \( \frac{m}{ABC, \ldots} = ABC, \ldots \); therefore, in that case, the general formula of No. 172 becomes \( \frac{1-v^t}{r} \frac{m}{ABC, \ldots} \); that is, the excess of the value of an annuity certain for the whole term \( r + t \), above that of an annuity on the whole duration of joint continuance of the last \( m \) surviving lives.

175. And if, in the case proposed in the last No. the annuity certain be a perpetuity, as in No. 173, the formula will become \( \frac{1}{r} \frac{m}{ABC, \ldots} \) the excess of the value of the perpetuity above the value of an annuity on the joint lives of the last \( m \) survivors; agreeably to No. 63.

176. Example 1. Required the present value of the absolute reversion of an estate in fee simple, after the extinction of the last survivor of three lives, \( A, B, C \), now aged 50, 55, and 60 years respectively; reckoning interest at 5 per cent.

The general Algebraical expression of this value has just been shown to be \( \frac{1}{r} \frac{m}{ABC} \).

But \( \frac{1}{r} = \frac{1}{0.5} = 20'000 \)

And \( \frac{m}{ABC} = 14'001 \) (68.)

Therefore 5'999 years' purchase is the value required. And if the annual produce of the estate, clear of all deductions, were L. 100, the title to the reversion would now be worth L. 599, 18s., agreeably to No. 76.

177. Ex. 2. An annuity for the term of 70 years certain (from this time), is to revert to \( Q \) and his heirs at the failure of a life \( A \), now 45 years of age; what is the present value of \( Q \)'s interest therein; reckoning the interest of money at 5 per cent.? In No. 174, the Algebraical expression of the required value is shown to be \( \frac{1-v^{70}}{r} - A \).

But \( \lambda v = x 1.05 = 1.9788107 \times 70 \)

\[ v^{70} = 0.32866 \times 2.5167490 \]

\[ 1-v^{70} = 0.967134 \]

\[ \frac{1-v^{70}}{r} = 0.967134 \times 0.05 = 19.34268 \]

Subtract \( A = 12.64800 \) (Tab. VI.)

remains 6.69468 years' purchase; so that if the annuity were L. 1000, the value of the reversion would be L. 6694, 13s. 7d.

178. Ex. 3. An annuity for the term of 70 years certain from this time, is to revert to Q and his heirs at the extinction of the survivor of two lives, A and B, now aged 40 and 50 years respectively; the interest of money being 5 per cent., it is required to determine the value of Q's interest in this annuity.

The algebraical expression of the value is,

\[ \frac{1-v^{70}}{r} - AB \] (174 and 171).

But by the last example

\[ \frac{1-v^{70}}{r} = 19.34268 \]

and by No. 66. \( AB = 15.06600 \)

So that the required value is 4.27668 years' purchase; and if the annuity be L. 1000, the present value of the reversion will be L. 4276, 13s. 7d.

IV. OF ASSURANCES ON LIVES.

179. Let the present value of the assurance (77 and 78) of L. 1 on the life of A be denoted by the Old English capital \( \mathcal{A} \), and that of an assurance on the joint continuance of any number of lives A, B, C, &c. by \( \mathcal{ABC} \), &c. Also, let the value of an assurance on the joint continuance of any m of them, out of the whole number \( m + \mu \) be denoted by \( \mathcal{ABC} \), &c.

180. And, in every case, let us designate the annual premium (83) for an assurance, by prefixing the character \( \circ \) to the symbol for the single premium; so that \( \circ \mathcal{A} \) may denote the annual premium for an assurance on the life of A; \( \circ \mathcal{ABC} \), &c. the same for an assurance on the joint continuance of all the lives, A, B, C, &c.; and \( \circ \mathcal{ABC} \), &c. the annual premium for an assurance on the joint continuance of the last m survivors out of the whole number \( m + \mu \) of those lives.

181. Then will \( \circ \mathcal{A} \) and \( \circ \mathcal{A} \), \( \circ \mathcal{ABC} \), &c. and \( \circ \mathcal{ABC} \), &c., \( \circ \mathcal{ABC} \), &c. and \( \circ \mathcal{ABC} \), &c. designate the single and annual premiums for assurances on the same life or lives for the term of t years only.

182. To determine \( \left( \frac{m}{t} \mathcal{ABC} \right) \) the present value of an assurance on the last m survivors out of \( m + \mu \) lives A, B, C, &c. for the term of t years only; that is, the present value of L. 1, to be received upon the joint continuance of these last m survivors failing in the term.

Solution.

By reasoning as in No. 79, it will be found, that a perpetuity, the first payment of which is to be made at the end of the year in which the last m survivors out of these \( m + \mu \) lives may fail in the term, will be of the same present value as \( \left( 1 + \frac{1}{r} \right) \frac{1}{1-v} \) pounds to be received in the same event (112 and 146); but, in No. 173, the value of the reversion of such a perpetuity in that event, was shown to be

\[ \frac{v}{1-v} \left[ 1 - \frac{m}{t} \mathcal{ABC} \right] = \frac{m}{t} \mathcal{ABC} \], &c.; whence it is manifest, that \( \frac{m}{t} \mathcal{ABC} \), &c.

\[ = \frac{v}{1-v} \left[ 1 - \frac{m}{t} \mathcal{ABC} \right] - \frac{m}{t} \mathcal{ABC} \], &c.

183. Since the annual premium for this assurance must be paid at the commencement of every year in the term, while the last m surviving lives all subsist (83); besides the premium paid down now, one must be paid at the expiration of every year in the term except the last, provided that these last m survivors all outlive it; but the present value of L. 1 to be received upon their surviving that last year is

\[ \frac{m}{t} \mathcal{ABC} \], &c., therefore all the future premiums are now worth \( \frac{m}{t} \mathcal{ABC} \), &c. \( \frac{m}{t} \mathcal{ABC} \), &c. \( \frac{m}{t} \mathcal{ABC} \), &c. \( \frac{m}{t} \mathcal{ABC} \), &c.

\[ v \left[ 1 - \frac{m}{t} \mathcal{ABC} \right] - \frac{m}{t} \mathcal{ABC} \], &c. = 1 - \frac{m}{t} \mathcal{ABC} \), &c.

\[ - (1-v) \left[ 1 - \frac{m}{t} \mathcal{ABC} \right] + \frac{m}{t} \mathcal{ABC} \], &c., whence we have

\[ \circ \mathcal{ABC} \], &c. = \frac{1}{1-v} \left[ 1 - \frac{m}{t} \mathcal{ABC} \right] + \frac{m}{t} \mathcal{ABC} \], &c.

\[ v = 1. \]

184. Corol. 1. When (t) the term of the assurance is not less than the greatest possible joint duration of any m of the proposed lives, \( \frac{m}{t} \mathcal{ABC} \), &c. = 0, \( \frac{m}{t} \mathcal{ABC} \), &c. = \( \mathcal{ABC} \), &c. and the general formulæ of the

185. Corol. 2. In the same manner it appears, that, for a single life, \( \bar{A} = v - (1-v) A \),

and \( \bar{A} = \frac{1}{1 + A} + v - 1 \)

186. Corol. 3. Also that \( \bar{A} = v(1 - t_a v^t) - (1-v) \bar{A} \),

or \( \bar{A} = v(1 - t_a v^t) - (1-v) \left( \frac{1}{1 + A} + v - 1 \right) \).

And \( \bar{A} = \frac{1 - t_a v^t}{1 - t_a v^t + \frac{1}{1 + A}} + v - 1 \)

that is, \( \bar{A} = \frac{1 - t_a v^t}{1 + A} + v - 1 \).

187. Corol. 4. When the assurance is on the joint continuance of all the lives, the formulae of No. 184 become respectively

\( \bar{ABC}, \&c. = v - (1-v) ABC, \&c. \)

and \( \bar{ABC}, \&c. = \frac{1}{1 + ABC, \&c.} + v - 1 \).

And those of numbers 182 and 183,

\( \bar{ABC}, \&c. = v \left( 1 - \frac{t(abc, \&c.)}{abc, \&c.} \right)^t - (1-v) \times \)

\[ \left[ ABC, \&c. - \frac{t(abc, \&c.)}{abc, \&c.} v^t \cdot t(ABC, \&c.) \right], \]

and \( \bar{ABC}, \&c. = \frac{1 - t(abc, \&c.)}{abc, \&c.} v^t \)

\[ 1 + ABC, \&c. - \frac{t(abc, \&c.)}{abc, \&c.} v^t \left[ 1 + t(ABC, \&c.) \right] + v - 1. \]

188. Corol. 5. According as the assurance is in the last survivor of two, or of three lives, the formulae of No. 184 become respectively

\( \bar{AB} = v - (1-v) \bar{AB}, \)

and \( \bar{AB} = \frac{1}{1 + \bar{AB}} + v - 1; \)

or \( \bar{ABC} = v - (1-v) \bar{ABC}, \)

and \( \bar{ABC} = \frac{1}{1 + \bar{ABC}} + v - 1. \)

And those of numbers 182 and 183 become

\( \bar{AB} = v \left[ 1 - \frac{t(ab)}{ab} \right]^t - (1-v) \bar{AB}, \)

and \( \bar{AB} = \frac{1 - t(ab)}{1 - t(ab) + \bar{AB}} + v - 1; \)

or \( \bar{ABC} = v \left[ 1 - \frac{t(abc)}{abc} \right]^t - (1-v) \bar{ABC}. \)

190. \( v \left[ 1 - \frac{m}{t(abc, \&c.)} \right]^t - (1-v) \bar{ABC, \&c.} \), the value of an assurance on any life or lives for the term of \( t \) years, which was given in No. 182, may also be expressed thus:

\[ \left[ 1 + \frac{m}{t(ABC, \&c.)} - \frac{m}{t(abc, \&c.)} \right] v - \frac{m}{t} \bar{ABC, \&c.} \]

And this, in words at length, is the rule given in No. 93.

191. When \( t \) is not less than the greatest possible joint duration of any \( m \) of the proposed lives, the last expression becomes \( \left( 1 + \frac{m}{ABC, \&c.} \right) v - \frac{m}{ABC, \&c.} \),

which is also equivalent to the first in No. 184; and, in words at length, is the rule given in No. 97, for determining the value of an assurance on any life or lives for their whole duration.

192. By substituting \( \frac{1}{1+r} \) for \( v \) (146) in the last expression, it becomes \( \frac{1 + ABC, \&c.}{1 + r} - \frac{ABC, \&c.}{r} \),

\[ = \frac{1}{1 + r} \bar{ABC, \&c.} \text{ or } \frac{1}{1 + r} \bar{ABC, \&c.} \]

And

Algebraical View.

\[ \frac{1}{r} - \text{ABC}, \&c. = \text{ABC}, \&c. \]

is the proposition enunciated in No. 81; \( \frac{1}{r} \) being the value of the perpetuity (112).

193. Examples of the determination of the single premiums for assurances, and of the derivation of the annual premiums from them, have been given in numbers 82—88, also in 95 and 96; but by the algebraical formulae given here, the annual premiums may be determined directly, without first finding the total present values of the assurances.

194. Example 1. Required the annual premium for an assurance on the life \( A \) now 50 years of age, interest 5 per cent.

According to No. 185, the operation is thus,

\[ \begin{align*} 1 + A &= 12.660 \lambda \\ \frac{1}{1 + A} &= 0.0789890 \times 2.8975663 \\ \text{adding } v &= 0.9523809, \text{ and subtracting unity,} \\ \text{we have } \circ A &= 0.0313699, \text{ agreeably to No. 85.} \end{align*} \]

195. Ex. 2. What should the annual premium be for an assurance on the last survivor of three lives \( A, B, C \), now aged 50, 55, and 60 years respectively, rate of interest 5 per cent?

Operation by No. 188.

\[ \begin{align*} (68) \quad 1 + \text{ABC} &= 15.001 \lambda \\ \frac{1}{1 + \text{ABC}} &= 0.0666622 \times 2.8238798 \\ v &= 0.9523809 \\ \circ \text{ABC} &= 0.0190431, \text{ agreeably to No. 88.} \end{align*} \]

No. 88.

196. Ex. 3. Required the annual premium for an assurance for 10 years only, on a life now 45 years of age, interest 5 per cent.

Operation according to No. 186.

\[ \begin{align*} v^{10} &= 6.18913 \lambda \times 1.7881068 \\ \frac{1}{v^{10}} &= 4073 \lambda \times 3.6099144 \\ a &= 4727 \times 4.3254144 \\ \frac{1}{v^{10}a} &= 5.28976 \lambda \times 1.7234356 \\ 1 + \frac{1}{v^{10}A} &= 11.347 \lambda \times 1.0548811 \\ \text{Subtract } 6.002 \lambda \times 0.7783167 \\ \text{from } 1 + A &= 13.648 \\ \text{remains } 7.646 \times 1.1165657 \\ 1 - \frac{1}{v^{10}A} &= 4.71024 \times 1.6730430 \\ \cdot 061604 \lambda \times 2.7896087 \\ v &= 0.952381 \\ \circ \text{A} &= 0.019885, \text{ agreeably to No. 96.} \end{align*} \]

What has been advanced from numbers 99 to 109, needs no algebraical illustration. (v.) ### TABLE I.

Showing the present Value of One Pound to be received at the End of any Number of Years not exceeding 50.

(See No. 9—12 of the preceding Article.)

| Years | 2 per Cent. | 2½ per Cent. | 3 per Cent. | 4 per Cent. | 5 per Cent. | 6 per Cent. | 7 per Cent. | 8 per Cent. | 9 per Cent. | |-------|-------------|--------------|-------------|-------------|-------------|-------------|-------------|-------------|-------------| | 1 | .980392 | .975610 | .970874 | .961538 | .952381 | .943396 | .934579 | .925926 | .917431 | | 2 | .961169 | .951814 | .942396 | .924556 | .907029 | .889996 | .873439 | .857339 | .841680 | | 3 | .942822 | .928599 | .915142 | .888996 | .863838 | .839619 | .816298 | .793832 | .772183 | | 4 | .923845 | .905951 | .888487 | .854804 | .822702 | .792094 | .762895 | .735030 | .708425 | | 5 | .905731 | .883854 | .862609 | .821927 | .785526 | .747258 | .712986 | .680583 | .649931 | | 6 | .887971 | .862297 | .837484 | .790315 | .746215 | .704961 | .663642 | .630170 | .596267 | | 7 | .870560 | .841265 | .813092 | .759918 | .710681 | .665057 | .622750 | .583490 | .547034 | | 8 | .853490 | .820747 | .789409 | .730690 | .676839 | .627412 | .582009 | .540269 | .501866 | | 9 | .836755 | .800723 | .766417 | .702587 | .644609 | .591898 | .543934 | .500249 | .460428 | | 10 | .820348 | .781198 | .744094 | .675564 | .619193 | .558395 | .508349 | .463193 | .422411 | | 11 | .804263 | .762145 | .722421 | .649581 | .584679 | .526788 | .475093 | .428888 | .387533 | | 12 | .788495 | .743556 | .701380 | .624597 | .556837 | .496969 | .444012 | .397114 | .355535 | | 13 | .773033 | .725420 | .680951 | .600574 | .530321 | .468839 | .414964 | .367698 | .326179 | | 14 | .757875 | .707727 | .661118 | .577475 | .505068 | .442301 | .387817 | .340461 | .299246 | | 15 | .743015 | .690466 | .641862 | .55265 | .481017 | .417265 | .362446 | .315242 | .274538 | | 16 | .728446 | .675625 | .623167 | .533908 | .458112 | .393646 | .338735 | .291890 | .251870 | | 17 | .714163 | .657195 | .605016 | .513973 | .436297 | .371364 | .316574 | .270259 | .231073 | | 18 | .700159 | .641166 | .587395 | .493628 | .415521 | .350444 | .295864 | .250249 | .211994 | | 19 | .686431 | .625528 | .570286 | .474642 | .405734 | .330513 | .276508 | .231712 | .194490 | | 20 | .672971 | .610271 | .553676 | .456387 | .376889 | .311805 | .258419 | .214548 | .178431 | | 21 | .659776 | .595386 | .537549 | .438834 | .358942 | .294155 | .241513 | .198656 | .163698 | | 22 | .646839 | .580865 | .521893 | .421955 | .341850 | .277505 | .225713 | .183941 | .150182 | | 23 | .634156 | .566697 | .506692 | .405726 | .325571 | .261797 | .210947 | .170315 | .137781 | | 24 | .621721 | .552875 | .491934 | .390121 | .310668 | .246979 | .197147 | .157699 | .126405 | | 25 | .609531 | .539391 | .477606 | .375117 | .295303 | .239999 | .184249 | .146018 | .115968 | | 26 | .597579 | .526235 | .463695 | .360689 | .281241 | .219810 | .172195 | .135202 | .106396 | | 27 | .585862 | .513400 | .450189 | .346817 | .267848 | .207368 | .160990 | .125187 | .097608 | | 28 | .574375 | .500878 | .437077 | .333477 | .255094 | .195630 | .150402 | .115914 | .089548 | | 29 | .563112 | .488661 | .424346 | .320651 | .242946 | .184557 | .140568 | .107328 | .082155 | | 30 | .552071 | .476743 | .411987 | .308319 | .231377 | .174110 | .131367 | .099377 | .075371 | | 31 | .541246 | .465115 | .399967 | .296460 | .220359 | .164255 | .122773 | .092016 | .069148 | | 32 | .530633 | .453771 | .388337 | .285058 | .209866 | .154957 | .114741 | .085200 | .063488 | | 33 | .520229 | .442703 | .377026 | .274094 | .199873 | .146186 | .107235 | .078889 | .058200 | | 34 | .510028 | .431905 | .366045 | .263552 | .190355 | .137912 | .100219 | .073045 | .053395 | | 35 | .500028 | .421371 | .355383 | .253415 | .181290 | .130105 | .093663 | .067635 | .048986 | | 36 | .490223 | .411094 | .345082 | .243669 | .172657 | .122741 | .087535 | .062625 | .044941 | | 37 | .480611 | .401067 | .334983 | .234297 | .164436 | .115793 | .081809 | .057986 | .041231 | | 38 | .471187 | .391285 | .325226 | .225285 | .156605 | .109239 | .076457 | .053690 | .037826 | | 39 | .461948 | .381741 | .315754 | .216621 | .149148 | .108056 | .071455 | .049713 | .034703 | | 40 | .452890 | .372431 | .306557 | .208289 | .142046 | .097222 | .066780 | .046031 | .031538 | | 41 | .444010 | .363347 | .297628 | .200278 | .135282 | .091719 | .062412 | .042621 | .029209 | | 42 | .435304 | .354485 | .288959 | .192375 | .128840 | .086527 | .058329 | .039464 | .026797 | | 43 | .426769 | .345899 | .280543 | .185168 | .122704 | .081630 | .054513 | .036541 | .024584 | | 44 | .418401 | .337404 | .272372 | .178046 | .116861 | .077009 | .050946 | .033834 | .022555 | | 45 | .410197 | .329174 | .264439 | .171198 | .111297 | .072650 | .047613 | .031528 | .020692 | | 46 | .402154 | .321146 | .256737 | .164614 | .105997 | .068838 | .044499 | .029007 | .018984 | | 47 | .394268 | .313313 | .249239 | .158233 | .100949 | .064658 | .041587 | .026859 | .017416 | | 48 | .386538 | .305671 | .241999 | .152195 | .096142 | .060998 | .038867 | .024869 | .015978 | | 49 | .378958 | .298216 | .234950 | .146341 | .091564 | .057546 | .036324 | .023027 | .014659 | | 50 | .371528 | .290942 | .228107 | .140713 | .087204 | .054288 | .033948 | .021321 | .013449 | ### TABLE II.

Showing the present Value of an Annuity of One Pound for any Number of Years not exceeding 50.

(No. 9.—12.)

| Years | 2 per Cent. | 3 per Cent. | 4 per Cent. | 5 per Cent. | 6 per Cent. | 7 per Cent. | 8 per Cent. | 9 per Cent. | |-------|-------------|-------------|-------------|-------------|-------------|-------------|-------------|-------------| | 1 | 9804 | 9756 | 9709 | 9615 | 9524 | 9434 | 9346 | 9259 | 9174 | | 2 | 19416 | 19274 | 19135 | 18861 | 18594 | 18334 | 18080 | 17833 | 17591 | | 3 | 28839 | 28560 | 28286 | 27751 | 27232 | 26730 | 26243 | 25771 | 25313 | | 4 | 38077 | 37620 | 37171 | 36299 | 35460 | 34651 | 33872 | 33121 | 32397 | | 5 | 47135 | 46458 | 45797 | 44518 | 43295 | 42124 | 41002 | 39927 | 38897 | | 6 | 56014 | 55081 | 54172 | 52421 | 50757 | 49173 | 47665 | 46229 | 44859 | | 7 | 64720 | 63494 | 62303 | 60021 | 57864 | 55824 | 53893 | 52064 | 50330 | | 8 | 73255 | 71701 | 70197 | 67327 | 64632 | 62098 | 59713 | 57466 | 55348 | | 9 | 81622 | 79709 | 77861 | 74353 | 71078 | 68017 | 65152 | 62469 | 59952 | | 10 | 89826 | 87521 | 85302 | 81109 | 77217 | 73601 | 70236 | 67101 | 64177 | | 11 | 97868 | 95142 | 92526 | 87605 | 83064 | 78869 | 74987 | 71390 | 68052 | | 12 | 105753 | 102578 | 99540 | 93851 | 88633 | 83938 | 79427 | 75361 | 71607 | | 13 | 113484 | 109832 | 106350 | 99856 | 93956 | 88527 | 83577 | 79038 | 74869 | | 14 | 121062 | 116909 | 112961 | 10531 | 98986 | 92950 | 87455 | 82442 | 77882 | | 15 | 128495 | 123814 | 119379 | 11184 | 10397 | 97192 | 91079 | 85595 | 80607 | | 16 | 135777 | 130550 | 125611 | 11657 | 10747 | 10165 | 95466 | 88514 | 81262 | | 17 | 142919 | 137122 | 131661 | 121657 | 112741 | 104773 | 97692 | 91216 | 85436 | | 18 | 149920 | 143584 | 137535 | 126598 | 116896 | 108276 | 100591 | 93719 | 87556 | | 19 | 156785 | 149789 | 143288 | 131339 | 120853 | 111581 | 103556 | 96036 | 89501 | | 20 | 163514 | 155892 | 148775 | 135903 | 124622 | 114699 | 105940 | 98181 | 91285 | | 21 | 170112 | 161845 | 154150 | 140292 | 128212 | 117641 | 108355 | 10168 | 92922 | | 22 | 176580 | 167654 | 159869 | 144511 | 131630 | 120416 | 110612 | 102007 | 94424 | | 23 | 182922 | 173321 | 164436 | 148568 | 134886 | 123034 | 112752 | 103711 | 95802 | | 24 | 189139 | 178850 | 169355 | 152470 | 137986 | 125504 | 114693 | 105288 | 97066 | | 25 | 195235 | 184244 | 174131 | 156221 | 140939 | 127834 | 116536 | 106748 | 98226 | | 26 | 201210 | 189506 | 178768 | 159892 | 143752 | 130062 | 118258 | 108100 | 99290 | | 27 | 207069 | 194640 | 183270 | 163296 | 146430 | 132015 | 119867 | 109352 | 100266 | | 28 | 212813 | 199640 | 187641 | 166631 | 148981 | 134062 | 121371 | 110511 | 101161 | | 29 | 218444 | 204535 | 191885 | 169837 | 151411 | 135907 | 122777 | 111534 | 101988 | | 30 | 223965 | 209303 | 196004 | 172920 | 153725 | 137648 | 124090 | 112578 | 102737 | | 31 | 229377 | 213954 | 200004 | 175885 | 155998 | 139291 | 125318 | 113498 | 103428 | | 32 | 234683 | 218492 | 203888 | 178736 | 158027 | 140840 | 126466 | 114350 | 104062 | | 33 | 239886 | 222919 | 207658 | 181476 | 160025 | 142302 | 127538 | 115139 | 104644 | | 34 | 244986 | 227238 | 211318 | 184112 | 161929 | 143681 | 128540 | 115869 | 105178 | | 35 | 249986 | 231452 | 214872 | 186646 | 163742 | 144982 | 129477 | 116546 | 105668 | | 36 | 254888 | 235563 | 218323 | 189083 | 165469 | 146210 | 130352 | 117172 | 106118 | | 37 | 259695 | 239573 | 221672 | 191426 | 167113 | 147368 | 131170 | 117752 | 106530 | | 38 | 264406 | 243486 | 224925 | 193679 | 168679 | 148460 | 131935 | 118289 | 106908 | | 39 | 269026 | 247303 | 228082 | 195845 | 170170 | 149491 | 132649 | 118786 | 107255 | | 40 | 273555 | 251028 | 231148 | 197928 | 171591 | 150463 | 133317 | 119246 | 107574 | | 41 | 277995 | 254661 | 234124 | 199931 | 172944 | 151380 | 133941 | 119672 | 107866 | | 42 | 282348 | 258206 | 237014 | 201856 | 174232 | 152245 | 134524 | 120067 | 108134 | | 43 | 286616 | 261664 | 239819 | 203708 | 175459 | 153062 | 135070 | 120432 | 108880 | | 44 | 290800 | 265038 | 242543 | 205488 | 176628 | 153892 | 135579 | 120771 | 108605 | | 45 | 294902 | 268330 | 245187 | 207200 | 177741 | 154558 | 136055 | 121084 | 108812 | | 46 | 298923 | 271542 | 247754 | 208847 | 178801 | 155244 | 136500 | 121374 | 109002 | | 47 | 302866 | 274675 | 250247 | 210429 | 179810 | 155890 | 136916 | 121643 | 109176 | | 48 | 306731 | 277732 | 252667 | 211951 | 180772 | 156500 | 137305 | 121891 | 109336 | | 49 | 310521 | 280714 | 255017 | 213415 | 181687 | 157076 | 137668 | 122192 | 109482 | | 50 | 314236 | 283623 | 257298 | 214822 | 182559 | 157619 | 138007 | 122335 | 109617 |

Perp. 50-0000 40-0000 33-3333 25-0000 20-0000 16-6667 14-2857 12-5000 11-1111 Perp. ## TABLE III

Showing the Sum to which One Pound will increase when improved at Compound Interest during any Number of Years not exceeding 50.

(No. 9—12.)

| Years | per Cent. | 2½ per Cent. | 3 per Cent. | 4 per Cent. | 5 per Cent. | 6 per Cent. | 7 per Cent. | 8 per Cent. | |-------|-----------|--------------|------------|-------------|-------------|-------------|-------------|-------------| | | | | | | | | | | | 1 | 1·02000 | 1·02500 | 1·03000 | 1·04000 | 1·05000 | 1·06000 | 1·07000 | 1·08000 | | 2 | 1·04040 | 1·05063 | 1·06090 | 1·08160 | 1·10250 | 1·12360 | 1·14490 | 1·16640 | | 3 | 1·06121 | 1·07689 | 1·09272 | 1·12484 | 1·15762 | 1·19101 | 1·22504 | 1·25971 | | 4 | 1·08243 | 1·10381 | 1·12559 | 1·16985 | 1·21556 | 1·26248 | 1·31079 | 1·36049 | | 5 | 1·10408 | 1·13141 | 1·15927 | 1·21665 | 1·27682 | 1·33826 | 1·39859 | 1·46032 | | 6 | 1·12616 | 1·15969 | 1·19402 | 1·26539 | 1·34009 | 1·41851 | 1·50073 | 1·58687 | | 7 | 1·14869 | 1·18869 | 1·22987 | 1·31593 | 1·40710 | 1·50360 | 1·60578 | 1·71382 | | 8 | 1·17166 | 1·21840 | 1·26670 | 1·36856 | 1·47745 | 1·59384 | 1·71816 | 1·85093 | | 9 | 1·19509 | 1·24886 | 1·30477 | 1·42312 | 1·55132 | 1·65949 | 1·83845 | 1·99905 | | 10 | 1·21899 | 1·28006 | 1·34391 | 1·48024 | 1·62889 | 1·79048 | 1·96715 | 2·15892 | | 11 | 1·24337 | 1·31209 | 1·38423 | 1·53945 | 1·71039 | 1·89829 | 2·10483 | 2·33163 | | 12 | 1·26824 | 1·34489 | 1·42576 | 1·60103 | 1·79535 | 2·01219 | 2·25219 | 2·51817 | | 13 | 1·29361 | 1·37851 | 1·46853 | 1·66504 | 1·88564 | 2·13292 | 2·40984 | 2·71963 | | 14 | 1·31948 | 1·41297 | 1·51259 | 1·73167 | 1·97992 | 2·26090 | 2·57853 | 2·93714 | | 15 | 1·34587 | 1·44830 | 1·55796 | 1·80994 | 2·07892 | 2·39655 | 2·75903 | 3·17216 | | 16 | 1·37279 | 1·48451 | 1·60470 | 1·89291 | 2·18287 | 2·54035 | 2·95216 | 3·42594 | | 17 | 1·40024 | 1·52162 | 1·65248 | 1·97490 | 2·29208 | 2·69277 | 3·15815 | 3·70001 | | 18 | 1·42825 | 1·55966 | 1·70213 | 2·05817 | 2·40661 | 2·85430 | 3·37993 | 3·96602 | | 19 | 1·45681 | 1·59865 | 1·75306 | 2·10869 | 2·52960 | 3·02560 | 3·61652 | 4·31570 | | 20 | 1·48595 | 1·63862 | 1·80611 | 2·11912 | 2·65329 | 3·20713 | 3·86968 | 4·66095 | | 21 | 1·51567 | 1·67958 | 1·86025 | 2·27876 | 2·78596 | 3·39954 | 4·14056 | 5·03384 | | 22 | 1·54598 | 1·72107 | 1·91603 | 2·36919 | 2·92591 | 3·60353 | 4·43040 | 5·43654 | | 23 | 1·57609 | 1·76461 | 1·97358 | 2·46716 | 3·07152 | 3·81975 | 4·74030 | 5·87146 | | 24 | 1·60844 | 1·80873 | 2·03279 | 2·56390 | 3·22510 | 4·04895 | 5·07296 | 6·34118 | | 25 | 1·64061 | 1·85394 | 2·09378 | 2·66586 | 3·38635 | 4·29171 | 5·34743 | 6·84847 | | 26 | 1·67342 | 1·90029 | 2·15650 | 2·77247 | 3·55607 | 4·54883 | 5·80733 | 7·39635 | | 27 | 1·70689 | 1·94750 | 2·22129 | 2·88369 | 3·73846 | 4·82246 | 6·21366 | 7·98601 | | 28 | 1·74102 | 1·99605 | 2·28792 | 2·99870 | 3·92612 | 5·11167 | 6·64883 | 8·62710 | | 29 | 1·77584 | 2·04641 | 2·35666 | 3·11861 | 4·11636 | 5·41838 | 7·11425 | 9·31727 | | 30 | 1·81136 | 2·09757 | 2·42726 | 3·24398 | 4·32194 | 5·74349 | 7·61225 | 10·06265 | | 31 | 1·84759 | 2·15001 | 2·50008 | 3·37313 | 4·53809 | 6·08101 | 8·14511 | 10·86679 | | 32 | 1·88454 | 2·20376 | 2·57508 | 3·50805 | 4·76494 | 6·43387 | 8·71527 | 11·73708 | | 33 | 1·92229 | 2·25858 | 2·65235 | 3·64381 | 5·00319 | 6·84059 | 9·32340 | 12·67605 | | 34 | 1·96068 | 2·31352 | 2·73190 | 3·79413 | 5·25384 | 7·25102 | 9·97811 | 13·69013 | | 35 | 1·99989 | 2·36721 | 2·81386 | 3·94608 | 5·51601 | 7·68608 | 10·67658 | 14·78534 | | 36 | 2·03989 | 2·43254 | 2·89528 | 4·10993 | 5·79181 | 8·14725 | 11·42994 | 15·96817 | | 37 | 2·08069 | 2·49355 | 2·98527 | 4·26809 | 6·08140 | 8·63608 | 12·22818 | 17·24562 | | 38 | 2·12290 | 2·55568 | 3·07478 | 4·43819 | 6·38547 | 9·15425 | 13·07927 | 18·62527 | | 39 | 2·16474 | 2·61957 | 3·16702 | 4·61636 | 6·70475 | 9·70350 | 13·99480 | 20·11529 | | 40 | 2·20804 | 2·68306 | 3·26205 | 4·80102 | 7·03989 | 10·28572 | 14·97445 | 21·72452 | | 41 | 2·25220 | 2·75219 | 3·35989 | 4·99361 | 7·39199 | 10·90286 | 16·02670 | 23·64288 | | 42 | 2·29724 | 2·82100 | 3·46069 | 5·19278 | 7·76158 | 11·55703 | 17·14425 | 25·33948 | | 43 | 2·34319 | 2·89152 | 3·56451 | 5·40049 | 8·14967 | 12·25045 | 18·34435 | 27·36640 | | 44 | 2·39005 | 2·96381 | 3·67145 | 5·61651 | 8·55715 | 12·98548 | 13·99480 | 29·55579 | | 45 | 2·43785 | 3·03790 | 3·78159 | 5·84117 | 8·98508 | 13·76461 | 14·90245 | 31·92044 | | 46 | 2·48661 | 3·11385 | 3·89504 | 6·07482 | 9·43425 | 14·59048 | 15·87262 | 34·47408 | | 47 | 2·53634 | 3·19170 | 4·01189 | 6·31786 | 9·90597 | 15·46591 | 16·87457 | 37·92302 | | 48 | 2·58707 | 3·27149 | 4·13225 | 6·57052 | 10·40127 | 16·93872 | 17·98907 | 40·21057 | | 49 | 2·63881 | 3·35328 | 4·25621 | 6·83384 | 10·92153 | 17·37750 | 18·92980 | 43·42741 | | 50 | 2·69159 | 3·43717 | 4·38390 | 7·10668 | 11·46740 | 18·42015 | 19·75702 | 46·90161 | ### TABLE IV

Showing the Amount to which One Pound per Annum will increase at Compound Interest in any Number of Years not exceeding 50.

(No. 9—12.)

| Years | 2 per Cent. | 2½ per Cent. | 3 per Cent. | 4 per Cent. | 5 per Cent. | 6 per Cent. | 7 per Cent. | |-------|-------------|--------------|-------------|-------------|-------------|-------------|-------------| | 1 | 1·0000 | 1·0000 | 1·000000 | 1·000000 | 1·000000 | 1·000000 | 1·000000 | | 2 | 2·0200 | 2·0250 | 2·030000 | 2·040000 | 2·050000 | 2·060000 | 2·070000 | | 3 | 3·0604 | 3·0756 | 3·090000 | 3·121600 | 3·152500 | 3·185600 | 3·214900 | | 4 | 4·1216 | 4·1525 | 4·188627 | 4·24644 | 4·310125 | 4·374616 | 4·449943 | | 5 | 5·2040 | 5·2563 | 5·309136 | 5·416325 | 5·525631 | 5·637093 | 5·750739 | | 6 | 6·3081 | 6·3877 | 6·468410 | 6·63975 | 6·801913 | 6·975319 | 7·153291 | | 7 | 7·4434 | 7·5474 | 7·662462 | 7·898294 | 8·142008 | 8·393888 | 8·654021 | | 8 | 8·5830 | 8·7961 | 8·992386 | 9·214226 | 9·549109 | 9·937468 | 10·259803 | | 9 | 9·7546 | 9·9545 | 10·159106 | 10·582795 | 11·026564 | 11·491316 | 11·977989 | | 10 | 10·9497 | 11·2034 | 11·463879 | 12·066107 | 12·577893 | 13·180795 | 13·816448 | | 11 | 12·1687 | 12·4835 | 12·807796 | 13·486351 | 14·206787 | 14·971643 | 15·783599 | | 12 | 13·4121 | 13·7956 | 14·192030 | 15·025805 | 15·917127 | 16·869941 | 17·888451 | | 13 | 14·6803 | 15·1404 | 15·617790 | 16·626838 | 17·712983 | 18·882198 | 20·140643 | | 14 | 15·9739 | 16·5190 | 17·086324 | 18·291911 | 19·598632 | 21·015066 | 22·550488 | | 15 | 17·2934 | 17·9319 | 18·598914 | 20·025588 | 21·578564 | 23·275970 | 25·129022 | | 16 | 18·6393 | 19·3802 | 20·156881 | 21·824531 | 23·657492 | 25·672528 | 27·888054 | | 17 | 20·0121 | 20·8647 | 21·761588 | 23·697512 | 25·640366 | 28·212880 | 30·840217 | | 18 | 21·4123 | 22·3863 | 23·414435 | 25·645413 | 28·192385 | 30·905653 | 33·999033 | | 19 | 22·8406 | 23·9460 | 25·116868 | 27·671229 | 30·539004 | 33·759992 | 37·879659 | | 20 | 24·2974 | 25·5447 | 26·870374 | 29·778079 | 33·065934 | 36·785591 | 40·995492 | | 21 | 25·7833 | 27·1833 | 28·676486 | 31·969202 | 35·719252 | 39·992727 | 44·865177 | | 22 | 27·2990 | 28·8629 | 30·36780 | 34·247970 | 38·505214 | 43·392290 | 49·005739 | | 23 | 28·8450 | 30·5844 | 32·452884 | 36·617889 | 41·430475 | 46·995828 | 53·436141 | | 24 | 30·4219 | 32·3190 | 34·426470 | 39·082604 | 44·501999 | 50·815577 | 58·176671 | | 25 | 32·0303 | 34·1578 | 36·459264 | 41·645908 | 47·727099 | 54·864512 | 63·249038 | | 26 | 33·6709 | 36·0117 | 38·553042 | 44·311745 | 51·113454 | 59·156383 | 68·674076 | | 27 | 35·3443 | 37·9120 | 40·709634 | 47·084214 | 54·669126 | 63·705766 | 74·483323 | | 28 | 37·0512 | 39·8598 | 42·930923 | 49·967583 | 58·402583 | 68·528112 | 80·697691 | | 29 | 38·7922 | 41·8363 | 45·218850 | 52·966286 | 62·322712 | 73·639798 | 87·463299 | | 30 | 40·5681 | 43·9027 | 47·575416 | 56·084938 | 66·438848 | 79·038186 | 94·460786 | | 31 | 42·3794 | 46·0003 | 50·002678 | 59·328335 | 70·760790 | 84·801677 | 102·073041 | | 32 | 44·2270 | 48·1503 | 52·502759 | 62·701469 | 75·298829 | 90·889778 | 110·218154 | | 33 | 46·1116 | 50·3540 | 55·077841 | 66·209527 | 80·663771 | 97·343165 | 118·933425 | | 34 | 48·0338 | 52·6129 | 57·730177 | 69·857909 | 85·666959 | 104·183755 | 128·258765 | | 35 | 49·9945 | 54·9282 | 60·462082 | 73·652225 | 90·320307 | 111·434780 | 138·236878 | | 36 | 51·9944 | 57·3014 | 63·275944 | 77·598314 | 95·856323 | 119·120867 | 148·913460 | | 37 | 54·0343 | 59·7339 | 66·174223 | 81·702246 | 101·628139 | 127·268119 | 160·337402 | | 38 | 56·1149 | 62·2273 | 69·159449 | 85·970836 | 107·709546 | 135·904206 | 172·561020 | | 39 | 58·2372 | 64·7830 | 72·234233 | 90·409150 | 114·095023 | 145·058458 | 185·640292 | | 40 | 60·4020 | 67·4026 | 75·401260 | 95·025516 | 120·799774 | 154·761966 | 199·635112 | | 41 | 62·6100 | 70·0876 | 78·603298 | 99·826536 | 127·839763 | 165·047684 | 214·609570 | | 42 | 64·8622 | 72·8398 | 82·023196 | 104·819598 | 135·231751 | 175·950545 | 230·632240 | | 43 | 67·1395 | 75·6608 | 85·483892 | 110·012382 | 142·993339 | 187·507577 | 247·776496 | | 44 | 69·5027 | 78·5523 | 89·048409 | 115·412877 | 151·143006 | 199·758032 | 266·120851 | | 45 | 71·8927 | 81·5161 | 92·719861 | 121·029392 | 159·700156 | 212·743514 | 285·749311 | | 46 | 74·3306 | 84·5540 | 96·501457 | 126·870568 | 168·685164 | 226·508125 | 306·751763 | | 47 | 76·8172 | 87·6679 | 100·39501 | 132·945390 | 178·119422 | 241·086612 | 329·224386 | | 48 | 79·3535 | 90·8596 | 104·408396 | 139·263206 | 188·025393 | 256·564529 | 353·270093 | | 49 | 81·9406 | 94·1311 | 108·540648 | 145·833734 | 198·426663 | 272·958401 | 378·999009 | | 50 | 84·5794 | 97·4843 | 112·796867 | 152·667084 | 209·347996 | 290·335905 | 406·528929 | ### TABLE V.

Exhibiting the Law of Mortality at Carlisle. (No. 32.)

| Age | Number who complete that Age | Number who die in the next interval | Age | Number who complete that Year | Number who die in their next Year | Age | Number who complete that Year | Number who die in their next Year | |-----|-----------------------------|------------------------------------|-----|-------------------------------|----------------------------------|-----|-------------------------------|----------------------------------| | 0 | 10,000 | 533 | 32 | 5528 | 56 | 69 | 2525 | 124 | | 1 Month | 9467 | 154 | 33 | 5472 | 55 | 70 | 2401 | 124 | | 2 Months | 9313 | 87 | 34 | 5417 | 55 | 71 | 2277 | 134 | | 3 | 9226 | 256 | 35 | 5362 | 55 | 72 | 2143 | 146 | | 6 | 8970 | 255 | 36 | 5307 | 56 | 73 | 1997 | 156 | | 9 | 8715 | 254 | 37 | 5251 | 57 | 74 | 1841 | 166 | | 1 Year | 8461 | 682 | 38 | 5194 | 58 | 75 | 1675 | 160 | | 2 Years | 7779 | 505 | 39 | 5136 | 61 | 76 | 1515 | 156 | | 3 | 7274 | 276 | 40 | 5075 | 66 | 77 | 1359 | 146 | | 4 | 6998 | 201 | 41 | 5009 | 69 | 78 | 1213 | 132 | | 5 | 6797 | 121 | 42 | 4940 | 71 | 79 | 1081 | 128 | | 6 | 6676 | 82 | 43 | 4869 | 71 | 80 | 953 | 116 | | 7 | 6594 | 58 | 44 | 4798 | 71 | 81 | 837 | 112 | | 8 | 6536 | 43 | 45 | 4727 | 70 | 82 | 725 | 102 | | 9 | 6493 | 33 | 46 | 4657 | 69 | 83 | 623 | 94 | | 10 | 6460 | 29 | 47 | 4588 | 67 | 84 | 529 | 84 | | 11 | 6431 | 31 | 48 | 4521 | 63 | 85 | 445 | 78 | | 12 | 6400 | 32 | 49 | 4458 | 61 | 86 | 367 | 71 | | 13 | 6368 | 33 | 50 | 4397 | 59 | 87 | 296 | 64 | | 14 | 6335 | 35 | 51 | 4338 | 62 | 88 | 232 | 51 | | 15 | 6300 | 39 | 52 | 4276 | 65 | 89 | 181 | 39 | | 16 | 6261 | 42 | 53 | 4211 | 68 | 90 | 142 | 37 | | 17 | 6219 | 43 | 54 | 4143 | 70 | 91 | 105 | 30 | | 18 | 6176 | 43 | 55 | 4073 | 73 | 92 | 75 | 21 | | 19 | 6133 | 43 | 56 | 4000 | 76 | 93 | 54 | 14 | | 20 | 6090 | 43 | 57 | 3924 | 82 | 94 | 40 | 10 | | 21 | 6047 | 42 | 58 | 3842 | 93 | 95 | 30 | 7 | | 22 | 6005 | 42 | 59 | 3769 | 106 | 96 | 23 | 5 | | 23 | 5963 | 42 | 60 | 3643 | 122 | 97 | 18 | 4 | | 24 | 5921 | 42 | 61 | 3521 | 126 | 98 | 14 | 3 | | 25 | 5879 | 43 | 62 | 3395 | 127 | 99 | 11 | 2 | | 26 | 5836 | 43 | 63 | 3268 | 125 | 100 | 9 | 2 | | 27 | 5793 | 45 | 64 | 3143 | 125 | 101 | 7 | 2 | | 28 | 5748 | 50 | 65 | 3018 | 124 | 102 | 5 | 2 | | 29 | 5698 | 56 | 66 | 2894 | 123 | 103 | 3 | 2 | | 30 | 5642 | 57 | 67 | 2771 | 123 | 104 | 1 | 1 | | 31 | 5585 | 57 | 68 | 2648 | 123 | | | |

### TABLE VI.

Showing the Value of an Annuity on a Single Life at every Age, according to the Carlisle Table of Mortality, when the rate of Interest is Five per Cent. (No. 65.)

| Age | Value | Age | Value | Age | Value | Age | Value | Age | Value | Age | Value | |-----|-------|-----|-------|-----|-------|-----|-------|-----|-------|-----|-------| | 0 | 12-083| 18 | 15-987| 35 | 14-127| 52 | 11-154| 69 | 6-643 | 86 | 2-830 | | 1 | 13-995| 19 | 15-901| 36 | 13-987| 53 | 10-892| 70 | 6-336 | 87 | 2-685 | | 2 | 14-983| 20 | 15-817| 37 | 13-843| 54 | 10-624| 71 | 6-015 | 88 | 2-597 | | 3 | 15-824| 21 | 15-726| 38 | 13-695| 55 | 10-347| 72 | 5-711 | 89 | 2-495 | | 4 | 16-271| 22 | 15-628| 39 | 13-542| 56 | 10-063| 73 | 5-435 | 90 | 2-399 | | 5 | 16-590| 23 | 15-525| 40 | 13-390| 57 | 9-771 | 74 | 5-190 | 91 | 2-321 | | 6 | 16-735| 24 | 15-417| 41 | 13-245| 58 | 9-478 | 75 | 4-989 | 92 | 2-412 | | 7 | 16-790| 25 | 15-303| 42 | 13-101| 59 | 9-199 | 76 | 4-792 | 93 | 2-518 | | 8 | 16-786| 26 | 15-187| 43 | 12-957| 60 | 8-940 | 77 | 4-609 | 94 | 2-669 | | 9 | 16-742| 27 | 15-065| 44 | 12-806| 61 | 8-712 | 78 | 4-422 | 95 | 2-596 | | 10 | 16-669| 28 | 14-942| 45 | 12-648| 62 | 8-487 | 79 | 4-210 | 96 | 2-555 | | 11 | 16-581| 29 | 14-827| 46 | 12-480| 63 | 8-258 | 80 | 4-015 | 97 | 2-428 | | 12 | 16-494| 30 | 14-723| 47 | 12-301| 64 | 8-016 | 81 | 3-799 | 98 | 2-278 | | 13 | 16-406| 31 | 14-617| 48 | 12-107| 65 | 7-765 | 82 | 3-606 | 99 | 2-045 | | 14 | 16-316| 32 | 14-506| 49 | 11-892| 66 | 7-503 | 83 | 3-406 | 100 | 1-624 | | 15 | 16-227| 33 | 14-387| 50 | 11-660| 67 | 7-227 | 84 | 3-211 | 101 | 1-192 | | 16 | 16-144| 34 | 14-260| 51 | 11-410| 68 | 6-941 | 85 | 3-009 | 102 | 0-753 | | 17 | 16-066| 35 | | | | | | | | 103 | 0-317 | ### TABLE VII.

Showing the Value of an Annuity on the Joint Continuance of Two Lives, according to the Carlisle Table of Mortality, when the difference of their Ages is Five Years, and the rate of Interest Five per Cent. (No. 65.)

| Ages | Value | |------|-------| | 0 & 5 | 10-551 | | 1 & 6 | 12-931 | | 2 & 7 | 13-258 | | 3 & 8 | 14-019 | | 4 & 9 | 14-402 | | 5 & 10 | 14-649 | | 6 & 11 | 14-731 | | 7 & 12 | 14-736 | | 8 & 13 | 14-689 | | 9 & 14 | 14-606 | | 10 & 15 | 14-500 | | 11 & 16 | 14-389 | | 12 & 17 | 14-284 | | 13 & 18 | 14-178 | | 14 & 19 | 14-069 | | 15 & 20 | 13-959 | | 16 & 21 | 13-853 | | 17 & 22 | 13-746 | | 18 & 23 | 13-636 | | 19 & 24 | 13-520 |

### TABLE VIII.

Showing the Value of an Annuity on the Joint Continuance of Two Lives, according to the Carlisle Table of Mortality, when the difference of their Ages is Ten Years, and the rate of Interest Five per Cent. (No. 65.)

| Ages | Value | |------|-------| | 0 & 10 | 10-649 | | 1 & 11 | 12-275 | | 2 & 12 | 13-087 | | 3 & 13 | 13-769 | | 4 & 14 | 14-106 | | 5 & 15 | 14-334 | | 6 & 16 | 14-419 | | 7 & 17 | 14-432 | | 8 & 18 | 14-395 | | 9 & 19 | 14-321 | | 10 & 20 | 14-221 | | 11 & 21 | 14-106 | | 12 & 22 | 13-987 | | 13 & 23 | 13-864 | | 14 & 24 | 13-797 | | 15 & 25 | 13-608 | | 16 & 26 | 13-483 | | 17 & 27 | 13-359 | | 18 & 28 | 13-255 | ANNULOSA.*

The Linnean arrangement of the Animal Kingdom has, with some slight emendations, been adopted as the ground-work of the zoological articles contained in the later editions of the Encyclopaedia. In these Supplemental volumes, we propose to introduce all the recent discoveries in Zoology, and also to avail ourselves of the opportunity thus afforded, of describing the various classes of animals under an arrangement more accordant to the improved views of Science, and to the Order of Nature. We shall have occasion fully to explain the principles upon which our system is founded, in the article ZOOLOGY. At present, it is only necessary to mention, that all the different branches of this grand department of Natural History will be treated of under these heads: viz. ANNULOSA, CIRRHIPEDAE, INSECTA, MOLLUSCA, RADIATA, and VERTEBROSA. From these heads, references will be made to the zoological articles in the body of the work, and also from the older names to the corresponding appellations in the arrangement adopted in these volumes.

The term ANNULOSA (from Annulus, a ring or segment), is applied to animals whose bodies are more or less divided transversely into segments.

**Class I.—CRUSTACEA.**

**History.**

All the Crustacea, as their name imports, are enveloped in a crust or shell (crusta). Many of the larger species were known to ancient Naturalists. They were named Crustacea by the Latins, Malacostracoi (Μαλακόστρακοι) by the Greeks. Aristotle has dedicated a chapter to the species known to him; Athenæus enumerates those used as food; and Hippocrates has made mention of such species as were considered useful in medicine.

To the observations of Aristotle, very little was added by Pliny; and from his time until that of Rondeletius, Belon, Gesner, Aldrovandus, and Johnson, who placed the Crustacea between the Fishes and Mollusca, little or nothing was done that tends in any way to elucidate the natural history or structure of these animals.

By the great reformer Linné, they were arranged under the genera Monoculus, Cancer, and Oniscus, along with apterous insects; but the most prejudiced of his followers now admit that they have characters sufficient to establish them as a distinct class.

This type of animals was proposed in one of the General Arrangements of the Annales de Muséum, by M. G. Cuvier. It comprehends five classes, the classification of which will form the subject of the present article.

As the leading characters of the classes are very obvious, we shall, in the first place, lay them before our readers through the medium of a table, and shall then detail them more fully.

- **Class 1. CRUSTACEA.** Branchiae or gills for respiration. Legs † for motion. - **Class 2. MYRIAPODA.** Tracheæ or air-tubes for respiration. Legs more than eight. Head distinct from the thorax. Antennæ two. - **Class 3. ARACHNIDES.** Tracheæ for respiration. Legs eight or six. Head not distinct from the thorax. Antennæ none. - **Class 4. INSECTA.** Tracheæ for respiration. Legs six. Head distinct from the thorax. Antennæ two. - **Class 5. VERMES.** Tracheæ for respiration. Legs none. Antennæ none.

J. C. Fabricius, a pupil of Linné, divided the Crustacea from insects, and formed several distinct classes for their reception; but as he has altered his system in his different works, it seems necessary only to state the last, which is given in the Supplement to his Entomologia Systematica. In this work is the following arrangement:

- **Class POLYGONATA.** Many maxillæ within the lip. Gen. 1. Oniscus, 2. Ligia, 3. Idotea, 4. Cymothoa, 5. Monoculus. - **Class KLEISTAGNATHA.** Many maxillæ, closing the mouth. Lip none. Gen. 1. Cancer, 2. Calappa, 3. Ocypode, 4. Leucosia, 5. Parthenope, 6. Inachus, 7. Dromia, 8. Dorippe, 9. Orithya, 10. Portunus, 11. Matutin, 12. Hippa, 13. Symethis, 14. Limulus. - **Class EXOCHINATA.** Many maxillæ outside the lip, covered by the palpi. Gen. 1. Albunea, 2. Scyllarus, 3. Pallinurus, 4. Palaeemon, 5. Alpheus, 6. Astacus, 7. Penaeus, 8. Cran-gon, 9. Pagurus, 10. Galathea, 11. Squilla, 12. Podsydon, 14. Gammarus.

Before the publication of this work, Müller (in

---

* The animals which compose this type are partly treated of in the Articles ENTOMOLOGY and HELMINTHOLOGY of the Encyclopaedia.

† For the comparative characters of this type, see Zoology. In the present article, we propose to give the characters and economy of the Genera, and an example of one Species of each, with the exception of INSECTA, which will be treated of under that head in a separate article.

‡ By this term, we mean those organs which actually perform the functions of legs. On this subject more will be said under the article Zoology.

|| This class was instituted by Dr Leach. Latreille comprehended the animals composing it under the general denomination ARACHNIDES. Crustacea. 1792) produced his celebrated work on the Entomostraca, which contains several crustaceous genera, which he disposed into the following groups.

Division I. Monoculi. With but one eye.

* Shell univalve. Gen. 1. Amymone, 2. Nauplius. (These two genera, Jurine of Geneva has discovered to be but larvae of the genus Cyclops of Müller.)

** Shell bivalve. Gen. 3. Cypris, 4. Cythere, 5. Daphnia.

*** Shell composed of a solid crust. Gen. 6. Cyclops, 7. Polyphemus.

Div. II. Binoculi. With two eyes.

* Shell univalve. Gen. 8. Argulus, 9. Caligus, 10. Limulus.

** Shell bivalve. Gen. 11. Lynceus.

A great portion of the species, composing the Entomostraca of Müller, were described by the microscopic observers, Joblot, Frish, Réaumur, De Geer, Baker, Ledermüller, and Geoffroy, and were by the latter author arranged into two genera, which he named Binoculus and Monoculus.

By that learned anatomist G. Cuvier, the Crustacea were considered as forming a class, which he placed between the Vermes and Insecta. This indefatigable observer discovered, that the Crustacea breathed by branchiae or gills, and he disposed them into three great sections, viz.

I. Monoculus, including the genera, 1. Limulus, 2. Caligus, 3. Cyclops, 4. Apus, 5. Polyphemus of Müller.

II. Cancer, comprehending the genera, 6. Cancer, 7. Inachus, 8. Astacus, 9. Palinurus, 10. Squilla, 11. Scyllarus of Fabricius:

III. Oniscus, including, 12. Physodes, 13. Omisus, 14. Cymothoa of Fabricius.

Lamarck, in his Système des Animaux sans Vertebres, has disposed the Crustacea into the two following orders, viz.

Ord. I. Crustace's pediociques, including the genera, *1. Cancer, 2. Calappa, 3. Ocypode, 4. Grapsus, 5. Dorippe, 6. Portunus, 7. Podophthalmus, 8. Matuta, 9. Maja, 10. Porcellana, 11. Leucosia, 12. Arctopsis, **13. Albunea, 14. Hippa, 15. Ranina, 16. Scyllarus, 17. Astacus, 18. Pagurus, 19. Galathea, 20. Palinurus, 21. Crangon, 22. Palemon, 23. Squilla, 24. Branchiopoda.

Ord. II. Crustace's sessiliociques, *25. Gammarus, 26. Asellus, 27. Caprella, 28. Oniscus, 29. Cyamus, 30. Ligia, 31. Cyclops, **32. Polyphemus, 33. Limulus, 34. Daphnia, 35. Amymone, 36. Cephaloculus. With the second section, Lamarck has placed the genus Forbicula (Lepisma saccharina of Linne), which, in our opinion, belongs to the genuine class Insecta.

By Duméril (Zoologie Analatique), these animals are placed in two orders, which are divided into families.

Ord. I. Entomostrace's.

Fam. 1. Aspidiota. Gen. 1. Limulus, 2. Calicus, 3. Binoculus, 4. Ozolus, 5. Apus.

Fam. 2. Ostracins. Gen. 6. Lynceus, 7. Daphnia, 8. Cypris, 9. Cythere.

Fam. 3. Gymnonesutes. Gen. 10. Argulus, 11. Crustacea Cyclops, 12. Polyphemus, 13. Zoe, 14. Branchiopoda.

Ord. II. Astacoides.

Fam. 4. Carcinoides. Gen. 15. Calappa, 16. Hepatus, 17. Dromia, 18. Cancer, 19. Matuta, 20. Portunus, 21. Podophthalmus, 22. Porcellana, 23. Ocypode, 24. Grapsus, 25. Pinnotheres.

Fam. 5. Oxyrinques. Gen. 26. Maja, 27. Leucosia, 28. Dorippe, 29. Orithyia, 30. Ranina.

Fam. 6. Macroures. Gen. 31. Pagurus, 32. Albunea, 33. Hippa, 34. Scyllarus, 35. Palinurus, 36. Galathea, 37. Astacus, 38. Penaeus, 39. Palemon, 40. Crangon.

Fam. 7. Arthrocephales. Gen. 41. Squilla, 42. Mysis, 43. Phronima, 44. Talitrus, 45. Gammarus.

The genera, 46. Oniscus, 47. Physodes, and 48. Armadillo, he has placed with apterous insects.

Bosc, in his Histoire Naturelle des Crustaces, has adopted, with some slight modifications, the system of Lamarck: he divides the Crustacea into two sections, viz.

Sec. I. Crustace's pediociques. Eyes pedunculated.

Div. 1. Body short; tail flat, simple, applied against the lower part of the abdomen.

Gen. 1. Cancer, 2. Calappa, 3. Ocypode, 4. Grapsus, 5. Dorippe, 6. Portunus, 7. Podophthalmus, 8. Orithyia, 9. Matuta, 10. Dromia, 11. Porcellana, 12. Leucosia, 13. Pinnotheres, 14. Maja.

Div. 2. Body oblong; tail elongated, terminated with appendices.

Gen. 15. Albunea, 16. Posydon, 17. Hippa, 18. Ranina, 19. Scyllarus, 20. Astacus, 21. Pagurus, 22. Galathea, 23. Palinurus, 24. Crangon, 25. Palemon, 26. Alpheus, 27. Penaeus, 28. Squilla, 29. Branchiopoda.

Sec. II. Crustace's sessiliociques. Eyes sessile.

Div. 1. Body covered with several crustaceous segments.

Gen. 1. Zoe, 2. Gammarus, 3. Talitrus, 4. Caprella, 5. Asellus, 6. Idotea, 7. Spharoma, 8. Ligia, 9. Cyamus, 10. Cymothoa, 11. Cyclops, 12. Bopyrus.

Div. 2. Body covered with a crustaceous shield, composed of one or two parts.

Gen. 30. Caligus, 31. Binoculus, 32. Limulus, 33. Apus, 34. Daphnia, 35. Cythere, 36. Cypris, 37. Polyphemus.

We shall now give the system of Latreille, published in his Considerations Générales.

Order I. Entomostracea.

Fam. 1. Aspidiota. Gen. 1. Limulus, 2. Apus, 3. Caligus, 4. Binoculus.

Fam. 2. Ostracoda. Gen. 5. Lynceus, 6. Daphnia, 7. Cypris, 8. Cythere.

Fam. 3. Gymnota. Gen. 9. Cyclops, 10. Polyphemus, 11. Zoe, 12. Branchiopoda.

Ord. II. Malacostracea.

Fam. 1. Cancrides. Gen. 13. Podophthalmus, 14. Portunus, 15. Dromia, 16. Cancer, 17. Hepatus, 18. Calappa, 19. Ocypode, 20. Grapsus, 21. Plagusia, 22. Pinnotheres. Class. CRUSTACEA.

Subclass I. ENTOMOSTRACA.

Tribe I. THECATA. Shell shield-shaped.

Fam. 1. XIPHOSURA. Gen. 1. Limulus.

Fam. 2. PNEUMONURA. Gen. 2. Caligus, 3. Binculus.

Fam. 3. PHYLOPODA. Gen. 4. Apus.

Tribe II. OSTRACODA. Shell bivalve.

Fam. 1. MONOPHTHALMA. Gen. 5. Lynceus, 6. Daphnia, 7. Cypris, 8. Cythera.

Tribe III. GYMNATA. Shell naked.

Fam. 1. PSEUDOPODA. Gen. 9. Cyclops.

Fam. 2. CEPHALOTA. Gen. 10. Polyphemus, 11. Zoe, 12. Branchiopoda.

Subclass II. MALACOSTRACA.

Order I. BRACHYURA.

Tribe 1. CANCERIDES. Gen. 13. Podophthalmus, 14. Lupa, 15. Portunus, 16. Carcinus, 17. Portunus, 18. Cancer, 19. Xantho, 20. Atelecyclus, 21. Dromia, 22. Hepatus, 23. Ocypode, 24. Uca, 25. Gonoplax, 26. Gecarcinus, 27. Grapsus, 28. Plagusia, 29. Pinnothoraces.

Tribe 2. OXYRINCHI. Gen. 30. Leucosia, 31. Maja, 32. Tharenopse, 33. Hyas, 34. Eurynome, 35. Blastus, 36. Pisa, 37. Inachus, 38. Macropodia (falsely written Septopodia), 39. Megalopa, 40. Corystes, 41. Mictyris, 42. Dorippe, 43. Orithya, 44. Matuta, 45. Ranina.

Order II. MACROURA.

Tribe 1. PAGURII. Gen. 46. Albunea, 47. Remipes, 48. Hippa, 49. Pagurus.

Tribe 2. PALINURI. Gen. 50. Scyllarus, 51. Pa-Crustacean, linurus, 52. Porcellana, 53. Galathea.

Tribe 3. ASTACINI. Gen. 54. Astacus, 55. Nephrops, 56. Thalassina, 57. Gebia (misprinted Upogebia), 58. Callianassa, 59. Alpheus, 60. Hippolyte, 61. Pandalus, 62. Penaeus, 63. Palinemon, 64. Athanas, 65. Crangon, 66. Mysis (repeated under the generic title Praunus).

Order III. GASTERURII.

Tribe 1. GNATHIDES. Gen. 67. Gnatha.

Tribe 2. GAMMERIDES. Gen. 68. Talitrus, 69. Orchestia, 70. Dexamine, 71. Leucothoe, 73. Melita, 74. Maera, 75. Gammarus, 76. Ampithoe, 77. Perusia, 78. Corophium, 79. Podocerus, 80. Jassa.

Tribe 3. PHRONIMARIDES. Gen. 81. Phronima.

Tribe 4. CAPRELLIDES. Gen. 82. Cyamus, 83. Caprella, 84. Proto.

Tribe 5. APSOEIDIES. Gen. 85. Apsoeides.

Tribe 6. ASELLIDES. Gen. 86. Anthura, 87. Campecopea, 88. Nessea, 89. Cymodoce, 90. Dynomenes, 91. Sphaeroma, 92. Cymothoa, 93. Limnoria, 94. Idotea, 95. Stenosomum, 96. Jadera, 97. Janira.

Tribe 7. ONISCIDES. Gen. 98. Ligia, 99. Philoscia, 100. Oniscus, 101. Porcellio, 102. Armadillo.

Lamarck, in his Extrait du Cours de Zoologie du Muséum d'Histoire Naturelle, &c. has given the following classification of these animals:

Order I. CRYPTOBRANCHA.

Section I. BRACHYURA.

* Body broader than long, rounded or truncated anteriorly.

CANCERIDES.

a Shore Crabs.

Genus 1. Cancer, 2. Dromia, 3. Hepatus, 4. Calappa, 5. Ocypode, 6. Grapsus, 7. Plagusia, 8. Pinnothoraces.

b Swimming Crabs.

Genus 9. Podophthalmus, 10. Portunus, 11. Matuta, 12. Orithya.

** Body subtriangular, terminated anteriorly in a point.

OXYRHYNCI.

Genus 13. Dorippe, 14. Leucosia, 15. Macropus, 16. Arctopsis, 17. Maja.

Section II. MACROURA.

* Tail furnished with cilia or hooks.

PAGURII.

Genus 18. Porcellana, 19. Corystes, 20. Ranina, 21. Albunea, 22. Hippa, 23. Pagurus.

** Tail furnished with swimming scales.

ASTACINI.

Genus 24. Scyllarus, 25. Palinurus, 26. Astacus, 27. Galathea, 28. Crangon, 29. Alpheus, 30. Palinemon.

Order II. GYMNORBRANCHA.

Section I. PEDIOCULLI.

SQUILLARI.

Genus 31. Squilla, 32. Mysis, 33. Branchiopoda.

Section II. SESIOCCULLI.

Genus 34. Caprella, 35. Phronima, 36. Gammarus, 37. Asellus, 38. Idotea, 39. Cymothoa, 40. Sphaeroma, 41. Ligia, 42. Oniscus, 43. Cyamus.

Section III. ENTOMOSTRACA.

1. With two eyes. Crustacea. Genus 44. Polyphemus, 45. Limulus, 46. Caligus, 47. Ozolus, 48. Zoea, 49. Lynceus.

2. With one eye. Genus 50. Daphnia, 51. Cytherea, 52. Cypris, 53. Cyclops, 54. Cephaloculus.

Having given a sketch of the systems adopted by these authors, we shall proceed to detail that proposed by Dr Leach in the Second Part of the eleventh Volume of the Transactions of the Linnean Society of London.

Classification.

Subclass I. Entomostacea. Legs branchial, or furnished with appendages. Mandibles wanting or simple. Eyes sessile or pedunculated.

Subclass II. Malacostraca. Legs simple, without appendages. Mandibles palpigerous. Eyes pedunculated or sessile.

Subclass I. Entomostacea.

The animals of this subclass are but little known, and consequently their arrangement is extremely imperfect. Some of the genera are parasitic, being found on the bodies of other animals, and some even undergo transformation during their growth.

The following arrangement is artificial; but is well calculated to enable the student to discover the genera.

Synopsis of the Genera.

Division I. Body covered by a horizontal shield. Eyes sessile.

Subdivision 1. Shield composed of two distinct parts. Gen. 1. Limulus.

Subdivision 2. Shield composed of but one part. * With jaws. Gen. 2. Apus. ** With a rostrum, but no jaws. a Antennae, four.

Gen. 3. Argulus. b Antennae, two.

4. Cecrops, 5. Caligus, 6. Pandarus, 7. An-thosoma.

Division II. Body covered by a bivalve shell. Eyes sessile.

Subdivision 1. Head porrected. Gen. 8. Lynceus, 9. Chydorus, 10. Daphnia.

Subdivision 2. Head concealed. Gen. 11. Cypris, 12. Cythere.

Division III. Body covered neither by a bivalve shell or shield. Eye one, sessile. Gen. 13. Cyclops, 14. Calanus, 15. Polyphemus.

Division IV. Body covered by neither a bivalve shell or shield. Eyes pedunculated. Gen. 16. Branchiopoda.

Division I. Subdivision 1. Gen. 1. Limulus, Muller, Fabr. &c. Xiphosura, Gronovius. Polyphemus, Lamarck.

Shell coriaceous, rounded in front, narrower behind; anterior shell largest, somewhat lunate, convex, with three carinae or keels; eyes two, ovate, very small, and scarcely prominent, one on each side of the lateral carina; hinder shell somewhat triangular, truncate-marginate, the sides toothed, having a moveable spine between each tooth; tail horny, three-sided, articulated to the notched extremity of the second shell by a hinge-like joint.

Antennae none.

Mandibles two, two-jointed, inserted under the anterior margin of the shell, their bases meeting; the second joint furnished with a moveable thumb-like process.

Legs ten, didactyle; fifth pair longest, the last joint but one with its extremity bearing elongate lamellae; anterior legs internally spinose near their base.

All the Limuli inhabit the sea. Monoculus Polyphemus of Linné belongs to the genus.

Limulus heterodactylus and L. viridescens of Latreille, probably form two distinct genera, belonging to the same subdivision.

Sp. 1. Sowerbii. Anterior shell with seven spines, arranged 1, 3, 3; hinder shell with five, 3, 1, 1, the lateral spines elongate and simple; tail above somewhat spinulose.

Limulus Sowerbii. Leach, Zool. Miscel. ii. 72. tab. 84.

The locality of this species, which is extremely common, is unknown.

Subdivision 2. * Gen. 2. Apus. Cuvier, Latreille, Leach. Apus, Scopoli.

Shell crustaceous-membranaceous, orbiculate-ovate, behind deeply emarginate; the back, with the exception of the anterior part, carinated; eyes two, inserted at the anterior and middle part of the back, somewhat prominent, slightly lunate, approaching each other, especially anteriorly where they touch each other.

Antennae two, short, somewhat filiform, biarticulated, scarcely exerted, inserted behind the mandibles.

Mandibulae two, cornuous, somewhat cylindric, short, hollow within, points arcuated and compressed, the extreme apex strait and very much denticulated.

Legs branchial and very numerous.

The Api inhabit stagnant waters and ponds; their anterior legs are spiny at the base, like those of the Limuli.

Sp. 1. Montagu. Carina of the shell produced into a point behind; anterior legs with articulated setæ; no lamella between the caudal setæ.

Plate XX. fig. 1. Apus Montagu, natural size; 2. anterior leg magnified; 3. part of one of the setæ of the anterior leg highly magnified; 4. setæ of the tail magnified.

Inhabits England, near Christ-church in Hampshire, where it was discovered by Montagu, who sent it to Dr Leach as the Linnean Monoculus apus.

Apus productus of Latreille, is synonymous with the Linnean Monoculus apus.

Subdivision 2. ** a. Gen. 3. Argulus, Muller, Jurine, Leach. Binoculus, Geoffroy, Latreille.

Shell oval, somewhat membranaceous, semi-transparent, anteriorly rounded, behind deeply notched; eyes two, hemispheric, inserted at the anterior and lateral parts of the clypeus.

Antennæ, very small, inserted above the eyes. Rostellum sterniform.

Legs twelve, unequal in size and form; first pair shorter, very membranaceous, capable of changing their form, broader at their tips, and formed for adhering to objects; second pair prehensile, curved, much thicker towards their base, the thighs furnished with three spinules beneath; tarsi three-jointed, the last joint with two claws and a pulvillus; four hinder pair inserted at the sides of the abdomen, somewhat cylindric, formed for swimming, with their points bifid.

Abdomen cylindric; tail bilobate.

Sp. 1. Follicaeus, Jurine.

Argulus delphinus. Müll. Entom. 123. Monoculus argulus. Fabr. Ent. Syst. ii. 489. Monoculus gyriini. Cuv. Tab. Elem. 454. Binoculus gasterostei. Latr. Gen. Crust. et Ins. Argulus foliaceus. Jurine, An. de Museum, vii. 451. Argulus argulus. Leach, Edin. Encycl. vii. 388.

This species, which is the only one of the genus that has hitherto been noticed, inhabits ponds and rivulets, adhering to the larvae of frogs and to fishes.

The larva has been described by Müller as a distinct species, under the name Argulus charon: in this state it differs from the full grown animal in size, and in having four cylindric, equal, biarticulated, penicillated oars, two of which are attached to the animal above the eyes, and are furnished at their tips with four setæ, the other two below the eyes being terminated with three setæ: The two anterior legs are incassated, elongated, and terminated by a strong bent claw.

The full grown animal lays from one to four hundred eggs, which are ovate and smooth, being generally deposited on stones in two contiguous longitudinal series. These eggs are hatched in about thirty days.

Subdivision 2. ** b.

Gen. 4. Cecrops. New Genus.

Shell coriaceous-membranaceous, composed of two parts; the anterior segment inverse heart-shaped, deeply and obtusely notched behind; anteriorly notched; the laciniae rounded and externally bearing the antennæ. Antennæ two-jointed, the first joint largest, thickest, the second bearing at its point a simple seta; hinder segment smaller, inversely heart-shaped, occupying the notch of the anterior segment.

Abdomen of the breadth of the shell, notched behind.

Rostrum elongate-conic, perpendicular, inserted between the middle and anterior legs; having on each side of its base a moveable ovate appendage.

Legs six; anterior pair biarticulated with a strong curved claw; second pair triarticulate, more slender, the last joint double, the exterior joint shortest; third pair strong, uniarticulate, with a very strong claw. Four legs spurious, placed behind the others, double, the double parts biarticulate, situated on a common coxa.

Female with two large ovate connected bags of a coriaceous substance, situated beneath the abdomen and projecting behind, in which she carries her eggs.

Sp. 1. Latreillii.

Plate XX. fig. 1, male; 2, female; 3, under side of male; 4, under side of female; 5, antennæ magnified, 6, anterior leg magnified; 7, hinder leg magnified; 8, middle leg.

Of this curious animal the history is unknown. That it is parasitic its structure evidently shows, and from analogy we may infer, that it is an inhabitant of the ocean, and that it attaches itself to the larger marine animals. There are several specimens preserved in the British Museum.

Gen. 5. Caligus, Müll. Latr. Bosc, &c.

Shell coriaceous-membranaceous, bipartite; the anterior segment inversely cordiform, very deeply notched behind (the notch receiving the hinder segment, which is round), the anterior part subproduced, notched; the laciniae at their base externally bearing antennæ; antennæ biarticulate, the first joint thickest, the second with a simple seta at its extremity.

Abdomen narrower than the thorax; with its base contracted and bearing the hinder legs, its extremity on each side with a rounded process of the length of the body. Rostrum rounded, rather more slender towards its apex, which is obtuse. Legs fourteen; anterior, second, and fourth pairs with a strong claw; the second pair short; the third slender elongate, the last joint double, with unequal laciniae; the fifth with the last joint on one side setose, the setæ ciliated on each side; the sixth with a double triarticulated tarsus, the last joints on each side setose, the setæ ciliated on each side; the seventh part with its last joint trifid.

The hinder segment of the thorax beneath, terminated by a large broad lamella, ciliated behind.

Caligus curtus of Müller, forms the type of this genus.

Sp. 1. Mulleri.

Plate XX. fig. 1, Natural size; 2, magnified; 3, seventh leg; 4, fifth leg; 5, one of the ciliated setæ of the fifth leg much magnified; 6, third leg; 7, fourth leg; 8, sixth leg.

Inhabits the common cod-fish. It was first sent to us by Dr Spalding, of Edinburgh, to whose kindness we are indebted for several very curious animals.

Gen. 6. Pandarus. New genus.

Caligus, Müller, Latr. Bosc, Leach, &c.

Shell coriaceous-membranaceous, composed of but one part, deeply notched behind; the angles acute; the middle of the notch toothed; anteriorly narrower, rounded, with a process on each side externally bearing the antennæ. Antennæ composed of two joints, the second joint terminated by several setæ. Abdomen somewhat narrower than the shell, the base above with two transverse lamellæ, the first of which is four-lobed, the second bilobate; the apex notched, with two filaments, longer than the body, with a lamella at their base above. Rostrum elongate, attenuated, inserted behind the anterior legs. Legs fourteen; anterior pair short, terminated by a short claw, and arising from beneath an ovate process; second pair with a double unequal tarsus; third pair without any determinate form, without any claw; fourth pair bifid; fifth and sixth pairs bifid, their coxae connected by a lamella; seventh pair bifid, the exterior lacinia longest, with a notch externally towards its apex.

Sp. 1. Bicolor. Shell and the middle of the abdominal lamellæ black; tail with filaments double the length of the body.

Inhabits the Synalus galus of Linné. Sp. 2. Boscii. Body entirely pale, testaceous; tail with filaments once and an half the length of the body. Inhabits the Squatius musculus of Linné.

Plate XX. fig. 1. Pandarus Boscii natural size; 2. magnified; 3. anterior leg magnified; 4. antenna magnified; 5. sixth leg magnified; 6. seventh leg magnified; 7. fifth leg magnified; 8. fourth leg magnified; 9. second leg magnified; 10. third leg magnified.

Gen. 7. Anthosoma. New genus.

Shell coriaceous-membranaceous unipartite, rounded before and behind; the anterior part as if unilobate, the lobe higher than the shell, behind on each side, bearing the antennae; antennae six-jointed. Abdomen much narrower than the shell, on every side imbricated with membranaceous foliaceous lamellae which surround or embrace it. Two of the lamellae are dorsal, the one being placed over the other; the other lamellae are placed on the sides of the belly, three on each side; apex of the abdomen terminated by two very long filaments, and with two shorter filaments below them. Rostrum elongate cylindric, inserted behind the anterior legs, furnished at its extremity with two stout cornaceous mandibles. Legs six; anterior pair three-jointed, the second joint near the apex above unidentate, the last terminated by a claw; second pair triarticulated, the last joint ovate, compressed; third pair biarticulate, the second joint very thick, internally dentated, armed at its extremity by a strong claw.

Sp. 1. Smithii. Plate XX. fig. 1. natural size; 2. two specimens adhering to part of the gill-cover of a shark; 3. antenna magnified; 4. middle leg magnified; 5. hinder leg magnified; 6. anterior leg magnified.

This species was discovered sticking to a shark which was thrown ashore on the coast of Exmouth in Devon, by T. Smith, Esq. of Paper Buildings, Temple. He informed the writer of this article, that it adhered solely by means of its anterior legs to the axillae and gill-covers, which were much thickened by the inflammation caused by their irritation.

Division II. Subdivision 1.

Gen. 8. Lynceus, Müll. Latr. Bosc, Leach.

Eyes two. Antennae four, capillary.

Sp. 1. Brachyurus. Shell globose; tail deflexed.

Lynceus brachyurus.

Müll. Entomost. 69. tab. 8. f. 1. 12. Bosc, Hist. Nat. des Crust. ii. 264. Leach, Edin. Encycl. vii. 388. Latr. Gen. Crust. et Ins. i. 17.

Monoculus brachyurus.

Fabr. Ent. Syst. ii. 407.

Inhabits marshes; is very common in the spring, moving about with great agility amongst aquatic plants. The female carries her eggs on the posterior and superior part of her belly.

Gen. 9. Chydorus.

Lynceus, Müll. Latr. Bosc, Leach.

Eyes two. Antennae two, capillary.

Sp. 1. Sphaericus. Shell globose; tail inflexed.

Lynceus sphericus.

Müll. Entomost. 71. t. ix. f. 7, 9. Latr. Gen. Crust. et Ins. i. 17.

Bosc, Hist. Nat. des Crust. ii. 264. Leach, Edin. Encycl. vii. 388.

Monoculus sphericus.

Fabr. Ent. Syst. ii. 497.

Gen. 10. Daphnia.

Müll. Latr. Bosc, Leach.

Eye one only. Antennae two, branching.

The extraordinary appearance presented by the animals of this genus caused them to be mentioned by Leuwenhoeck, Needham, Swammerdam, and other microscopical investigators. Their shell, although apparently bivalve, is formed but of one piece, open in front. Their head is terminated by a kind of pointed but immovable beak. Their mouth is placed within the shell. The eye is absolutely single, and not formed by the union of two eyes, and is covered by granules. The number of legs is not known. One of the species is called Pulex caudatus by Scheffer, 1755, t. 1. f. 1. 8. and in the Encyclopaedia Britannica, Pl. 37. fig. 1, a species is rudely figured as an animalcule.

Sp. 1. Pulex. Tail inflexed; shell mucronated behind.

Monoculus pulex.

Linn. Faun. Suec. 2047. Fabr. Ent. Syst. ii. 491.

Daphnia pennata.

Müll. Entom. 82. t. 12. f. 4. 7.

Daphnia pulex.

Latr. Gen. Crust. et Ins. i. 18. Leach, Edin. Encycl. vii. 338.

Inhabits ponds and marshes. Geoffroy (Hist. des Insect. ii. 655.) has given a description of this species under the very expressive name of Monocle le Perroquet d'eau.

Subdivision 2.

Gen. 11. Cypris, Müll. Latr. Bosc, Leach.

Antennae terminated by a brush.

Many of the Cyprides were noticed by Joblot, Ledermuller, Baker, and De Geer, but they were first reduced to one genus by the illustrious Müller.

The animals of this genus inhabit pools and ditches, containing pure water; they swim with very great rapidity, and, whilst in motion, conceal their whole body within their shell, which is truly bivalve. The members of the body of the cyprides are not known; they move with such quickness whilst living, and are so soft when dead, that it is scarcely possible to investigate their characters more fully. Their antennae are long, and very flexible; these organs are furnished at their tips with a brush or pencil, composed of hairs. The hinder part of their bodies is formed of a tail, which is almost entirely concealed within their shell. Their head is terminated by an elongate point, and they have but one eye. Of their economy nothing is known, but they have been observed to change their covering like other Entomostraca.

Plate XX. fig. 1. Cypris nephroides natural size; 2. magnified.

Sp. 1. Conchacea. Shell ovate, tomentose.

Monoculus conchaceus.

Linn. Faun. Suec. 2050. Fabr. Ent. Syst. ii. 496.

Cypris pubera.

Müll. Entomost. 56. t. 5. f. 1. 5. Cypris conchacea.

*Latr. Gen. Crust. et Insect. i. 18.* *Leach, Edin. Encycl. vii. 388.*

Inhabits France, Germany, and England. *Gen. 12. Cytherea.* *Müll. Latr. Bosc, Leach.* Antennae simply pilose.

This genus was first discovered and established by Müller, who first observed all the species described in his Entomostraca. It is distinguished from *Cypris* by the antennae, which are not terminated by a pencil of hairs. The legs are eight in number, and are rarely drawn within the shell, which is really bivalve.

The Cytheres have no tail, and their antennae, like those of the Cyprides, have their articulations pilose. They have but one eye. All the species inhabit the sea, and may be found among the conservae and corallines, which fill the pools left by the tide in most of the rocky coasts of Europe.

Sp. 1. *Viridis.* Shell reniform, velvety, and green. *Cythere viridis,* *Müll. Entomost. 64, tab. 7, f. 1, 2.* *Latr. Gen. Crust. et Insect. i. 19.* *Bosc, Hist. Nat. des Crust. ii. 261.* *Leach, Edin. Encycl. vii. 388.*

Inhabits the European Ocean. Is occasionally found on the shores of Scotland amongst fuci and conservae.

Division III.

*Gen. 13. Cyclops,* *Müll. Lam. Latr. Bosc, Leach.* Body ovate-conic, elongate. Eye one, situated on the thorax. Antennae four, simple. Legs eight.

All the animals of this genus inhabit fresh waters. The females carry their eggs in a pouch resembling a bunch of grapes on each side of the tail. The organs of generation of the male are placed in the antennae; those of the female beneath the belly, at the base of the tail, which is abruptly narrower than the abdomen. The antennae are hairy at the base of their joints.

Sp. 1. *Geoffroyii.* Tail strait and bifid; colour brownish.

Monoculus quadricornis.

*Linn. Fn. Syst. 2049.* *Fabr. Entom. Syst. ii. 500.*

Cyclops quadricornis.

*Müll. Entomost. 109, t. 18, f. 1, 14.* *Latr. Gen. Crust. et Insect. i. 19.* *Bosc, Hist. Nat. des Crust. ii. 228, pl. 18, f. 4.* female.

*Leach, Edin. Encycl. vii. 388.*

Inhabits Europe. Is very common in fresh waters.

Geoffroy, in his *Histoire Naturelle des Insectes* (tom. ii. 656, pl. 21, f. 5.), has described and figured this species under the title *Le Monocle à queue fourchue.* It was first noticed by Leuwenhoeck, and his observations were afterwards increased by those of Baker, Roesel, and De Geer.

*Gen. 14. Calanus.* *Cyclops,* *Müll. Latr. Bosc, Leach.* Body ovate conic, elongate. Eye one, situated on the thorax. Antennae two, simple. Legs eight.

*Cyclops longicornis.* *Müll. Entomost. 115, t. 19, f. 7, 9.* *Latr. Gen. Crust. et Insect. i. 20.* *Bosc, Hist. Nat. des Crust. ii. 229.* *Leach, Edin. Encycl. vii. 389.*

Monoculus longicornis.

*Fabr. Ent. Syst. ii. 501.*

Inhabits the Norwegian and German Ocean. It was first discovered by the celebrated Gunnerus in the Finnmarkian Sea, and was described by Müller in his *Zoologie Danica Prodromus,* under the title Cyclops Finnarchicus, which name he afterwards changed in his *Entomostraca.*

*Gen. 15. Polyphemus,* *Müll. Latr. Bosc, Leach.* *Cephaloculus,* *Lamarck.* Eye one, forming the head; legs ten; two bifid, elongate, and extended horizontally.

This genus, containing one species, was established by Müller. Naturalists have generally considered it as the larva of some other genus, but the multiplied observations of Müller, Lamarck, and Bosc, have cleared up all doubts on the subject, and it is now generally admitted as a distinct and perfect animal. The head is round, distinct from the thorax, and apparently composed of one large eye. The body is divided into two parts by a kind of contraction; the anterior part contains the legs and tail; the posterior part contains only the eggs and the young. The tail is long, and is attached to the under part of the body, being terminated by two setae.

The *polyphemus* inhabits marshes and moves quickly by the combined efforts of its short legs and bifid oars, generally swimming on its back.

Sp. 1. *Oculus.* Body luteous, with a few blue spots.

Polyphemus oculus.

*Müll. Entom. 119, pl. 20, f. 1, 5.* *Latr. Gen. Crust. et Insect. i. 20.* *Bosc, Hist. Nat. des Crust. ii. 290, pl. 18, f. 5, 6.* *Leach, Edin. Encycl. vii. 389.*

Cephaloculus stagnorum.

*Lam. Syst. des Animaux sans Vert. 170.*

Inhabits lakes and marshes. In the *Edinburgh Encyclopaedia,* Dr Leach stated it as probable that there were several species that had been confounded with *P. oculus,* but repeated observations have since convinced him of the error of this conjecture, and led to the conclusion that there is but one, which is subject to very considerable variation in size and colour.

Division IV.

*Gen. 16. Branchiopoda.* *Lam. Latr. Bosc, Leach.* Body filiform and very soft. Head divided from the thorax by a very narrow but distinct neck. Eyes two, lateral. Antennae two, short, two-jointed, capillary inserted behind and above the eyes. Front with two moveable processes (which are broader towards the apex in the male sex), that are notched, those of the female furnished with a papilla at their point.

Subsidiary Observations. In the front of the male, at the base of the moveable processes on the front, are two long hair-like filaments, and the clypeus is, as it were, double. In both sexes, the mouth is armed with a hooked rostriform papilla, supported by four little processes. The trunk of the body is keel-shaped, and is formed of eleven joints, each of which is furnished with two branchial feet, the anterior pair with two, the posterior pairs with three lamellae. The tail is about the length of the body, composed of Crustacea, six or nine obscure joints, the oval segment bearing two fins.

The organs of generation are situated at the base of the tail.

Sp. 1. Stagnalis. Body transparent, of a light brown colour, slightly tinged with green or blue, particularly on the head and legs.

Cancer stagnalis. Linn. Fa. S. 2043.

Branchiopoda stagnalis.

Lam. Syst. des Anim. sans Vert. 161. Latr. Gen. Crust. et Ins. i. 22. Leach, Edin. Encycl. vii. 389.

We shall transcribe the ingenious observations made on this species by the late Dr Shaw of the British Museum, published in the first volume of the Transactions of the Linnaean Society of London.

"It is generally found in such waters as are of a soft nature, and particularly in those shallows of rain-water which are so frequently seen in the spring and autumn, and in which the Monoculus pulex of Linnaeus, and other small animals, abound. At first sight, it bears some resemblance to the larva of a dytiscus; but, when viewed closely, it is found to be of a much more curious and elegant appearance than that animal. The legs, of which there are several pair (eleven?) on each side, are flat and filmy, and have the appearance of so many waving fins, of the most delicate structure imaginable. The whole animal is extremely transparent, and the general colour is brown, slightly tinged with blueish-green.

"Monoculus conchaceus" of Linnaeus very frequently assaults them, and adheres with such force to their tails and legs, as sometimes to tear off a part in the struggle. It delights much in sunshine, during which it appears near the surface of the water, swimming on its back, and moving in various directions, by the successive undulations of its numerous fin-like legs, and moving its tail in the manner of a rudder. On the least disturbance, it starts in the manner of a small fish, and endeavours to secret itself, by diving in the soft mud. It changes its skin at certain periods, as is evident from the exuviae or shoughs being frequently found in the water in which these animals are kept.

"In March and April, the females deposit their eggs without any settled order, and perfectly loose in the water. They appear to the naked eye, like very minute globules of a light brown colour. Each ovum, when magnified, closely resembles the farina of a mallow. It is thickly beset with spines on every side, and coated over with a transparent gelatinous substance, reaching just to the extremities of the spines, and is most probably intended to assist in causing them to adhere to the substances on which they may chance to fall, or as a security from the attacks of smaller animals. In about a fortnight or three weeks, the eggs are hatched, and the young animals may be seen to swim with great liveliness, by means of three very long pair of arms or rowers, which appear disproportionate to the size of the animal, and indeed it bears, in this very small state, not much resemblance to the form it afterwards assumes; but, in the short space of a very few hours, the body assumes a lengthened form, and begins to acquire the tail-fin. The eyes in this state do not appear pedunculated. On the seventh day after hatching, they approach pretty nearly the form of the perfect animal; they, however, still retain the two first pairs of rowers or arms. The legs are at this period very visible. About the ninth day it loses the long ears, and appears still more like the animal in its advanced state."

The Doctor farther observes, that it is highly probable, that a considerable time elapses before the animal assumes its full size, but the time he was unable to determine, as those he kept died before they had acquired any considerable size. When first hatched, they are scarcely larger than the common mite.

Subclass II. MALACOSTRACA.

This subclass has occupied a very considerable portion of attention, the result of which shall be given in the following pages.

Legion I. PODOPHTHALMA. Eyes pedunculated or elevated on footstalks.

Order I. BRACHYURA. Tail short and simple at its extremity.

Order II. MACROURA. Tail lengthened with appendices at its extremity.

Legion II. EDRIOPHTHALMA. Eyes sessile.

Legion I. PODOPHTHALMA.

The Malacostraca podophthalma include those animals, which, in common language, are denominated Crabs, Lobsters, Cray-fish, Prawns, Pandals, and Shrimps, all of which have the power of reproducing their claws when they are lost.

Crabs and Lobsters are said "to change their crust annually," but this, like many other statements in zoology, is incorrect. We have seen Maja squinado and the common Lobster (Astacus gammarus) so overgrown with Ostreae, Anomiae, Flustra, Sponge, and Sertulariae, as scarcely to be enabled to move the joints of their legs, and as the oysters were of a year's growth; and as the anomiae were above them, and the coralline matters upon the anomiae, there can be no doubt but the crust was at least two years old.

Order I. BRACHYURA.

Latreille arranges the Brachyura (from the proportional breadth and length of the thorax or shell) into two families; but the discovery of genera, unknown to that illustrious Entomologist, has convinced us that such a distribution is extremely unnatural; and although, from the infant state of our knowledge, we cannot venture to propose named divisions; yet we shall endeavour to dispose the genera into what appear to be natural groupes.

Synopsis and distribution of the Genera.

A. Abdomen of the male five-jointed, the middle joint longest; of the female seven-jointed.

Anterior pair of legs didactyle.

Division I. Shell nearly rhomboidal. Two anterior legs very long, with deflexed fingers.

Genus I. LAMBRUS.

Division II. Shell truncate behind. Two anterior legs of the male elongate, of the female moderate. Subdivision 1. Antennae long, ciliated on each side.

Genus 2. Corystes.

3. Thia.

4. Atelecyclus.

Subdivision 2. Antennae moderate, simple. Hinder pair of legs with compressed claws.

Genus 5. Portumnus.

6. Carcinus.

7. Portunus.

8. Lupa.

Subdivision 3. Antennae moderate, simple. Four hinder pair of legs compressed.

Genus 9. Matuta.

Subdivision 4. Antennae simple, short. Four hinder pair of legs simple.

Genus 10. Cancer.

11. Xantho.

12. Calappa.

B. Abdomen in both sexes seven-jointed. Two anterior legs didactyle.

Division III. Eight hinder legs simple, and alike in form.

Subdivision 1. Shell anteriorly arcuated, the sides converging to an angle.

(Two anterior legs unequal).

Genus 13. Pilumnus.

14. Gecarcinus.

Subdivision 2. Shell quadrate or subquadrate. Eyes inserted in the front.

* Shell quadrate. Eyes with a short peduncle.

Genus 15. Pinnoteres.

** Shell quadrate. Eyes with a long peduncle.

Genus 16. Octopode.

17. Uca.

18. Gomoplax.

Subdivision 3. Shell quadrate. Eyes inserted at the anterior angles of shell.

Genus 19. Grapsus.

Division IV. Two hinder legs at least dorsal.

Subdivision 1. Two posterior legs dorsal. Eyes with the first joint of the peduncle elongated.

Genus 20. Homola.

Subdivision 2. Four hinder legs dorsal. Eyes with the first joint of the peduncle short.

Genus 21. Dorippe.

22. Dromia.

Division V. Shell rostrated in front. Eight hinder legs alike and simple.

Subdivision 1. Fingers deflexed.

Genus 23. Eurynome.

24. Parthenope.

Subdivision 2. Fingers not deflexed. External antennae with the first joint simple. Anterior pair of legs distinctly thicker than the rest.

Genus 25. Pisa.

26. Lissa.

Subdivision 3. Fingers not deflexed. External antennae with their first joint simple. Anterior pair of legs scarcely thicker than the others, which are moderately long.

Genus 47. Maja.

Subdivision 4. Fingers not deflexed. External antennae with the first joint simple. Anterior pair of legs about the thickness of the rest, which are very long and slender.

Genus 28. Egeria.

29. Doclea.

Subdivision 5. Fingers not deflexed. External antennae with the first joint externally dilated.

Genus 30. Hyas.

C. Abdomen in both sexes six-jointed. Two anterior legs didactyle.

Division VI. Fifth pair of legs minute, spurious.

Genus 31. Lithodes.

Division VII. Second, third, fourth, and fifth pair of legs alike and slender.

Subdivision 1. Eyes retractile.

Genus 32. Inachus.

Subdivision 2. Eyes not retractile.

Genus 33. Macropodia.

D. Abdomen of the male six-jointed; of the female five-jointed; the last joint very large. Eyes not retractile.

Division VIII.

Genus 34. Leptopodia.

35. Pactolus.

E. Abdomen of both sexes four-jointed. Two anterior legs didactyle.

Division IX.

Genus 36. Leucosia.

37. Ixa.

The following genera belong to this order, but their situation has not yet been fixed, namely,

Genus 38. Hepatus.

39. Plagusia.

40. Mictyris.

41. Orithya.

42. Ranina.

43. Megalopa.

A. Division I.

Gen. 1. Lambrus. Leach.

External antennae simple. External double palpi, with the second joint of their internal footstalk, internally notched for the insertion of the palpi.

Sp. 1. Longimanus. Thorax spiny, the spines simple; arms smooth beneath.

Parthenope longimana. Fabr. Ent. Syst. Supl. 333.

Maja longimana.

Bosc, Hist. Nat. des Crust. i. 250. pl. 7. f. 1.

Lambrus longimanus.

Leach, Trans. Linn. Soc. xi. 310.

This species inhabits the Indian Ocean.

Division II. Subdivision 1.

Gen. 2. Corystes, Latr. Leach.

External antennae longer than the body, the third segment composed of elongate, cylindric joints. External double palpi with the internal footstalk narrow, the second joint largest, having its internal side broadly emarginate. Anterior pair of legs; of the male twice the length of the body, subcylindric, the hand gradually somewhat thicker and somewhat compressed; of the female of the length of the body, with a compressed hand; other legs with tibiae and tarsi of equal length; claws elongate, strait, acute and longitudinally sulcated. Abdomen of the male with the first joint linear transverse, the second longer and produced on each side, third nearly equally quadrate, the fourth transverse and narrower than the third, the fifth narrower, nearly triangular, with the tip rounded; of the female with six first joints transverse ar- Cassivelaunus. Shell granulated, crenulated behind; front bifid; the sides tridentate.

Cancer cassivelaunus.

Penn. Brit. Zool. iv. 6. t. 7. male and female. Herbst, i. 195. t. 12. f. 72. male.

Cancer personatus. Herbst, i. 193. t. 12. f. 71. female.

Albunea dentata. Fabr. Supl. Ent. Syst. 398.

Boeck, Hist. Nat. des Crust. ii. 4.

Corystes dentatus. Latr. Gen. Crust. et Ins. i. 40.

Corystes cassivelaunus.

Leach, Edinb. Encycl. vii. 395.

Trans. Linn. Soc. xi. 312.

C. cassivelaunus inhabits most of the sandy shores of the European Ocean, and are often thrown up after heavy gales of wind. Latreille formerly considered the male as a distinct species, under the title Corystes longimanus.

Gen. 3. Thia. Leach.

External antennae longer than the body, the third segment composed of elongate cylindric joints. External double palpi with the second joint of their internal footstalk much shorter than the first, with its internal apex truncate-emarginate. Anterior pair of legs of the male a little longer than the body, with the hand compressed; other legs with the tarsi half the length of the tibia, with acute flexuous claws, which are longitudinally sulcated. Abdomen of the male with the first joint transverse, arcuate, linear; the second a little longer, anteriorly slightly arcuatedly produced; the third very long, narrower towards the apex, which is slightly emarginate; the fourth subquadrate, apex slightly notched; the fifth triangular. Shell somewhat circular with the sides gradually converging into an angle behind; hinder part somewhat granulate-margined; the front somewhat produced. Eyes very small, scarcely prominent; orbit without any fissure behind.

Sp. 1. Polita. Shell convex, polished and sprinkled with punctures; orbit behind emarginate; sides on each side obscurely four-folded; front entire and arcuate.

Cancer residuus. Herbst, iii. 59. t. 48. f. 1.

This polita. Leach, Trans. Linn. Soc. xi. 312.

Locality unknown.

Gen. 4. Atelecyclus. Leach.

External antennae half the length of the body, the third segment composed of elongate and cylindric joints. External double palpi with the second joint of the internal footstalk shortest, with the internal apex produced, and the internal side notched towards the joint. Anterior legs of the male longer than the body, with a compressed hand; of the female as long as the body, with a compressed hand; other legs with tibiae and tarsi of equal lengths, furnished with elongate, quadrate nails, that are longitudinally sulcated, having their tips naked, rounded and sharp, the hinder ones obscurely subcompressed. Abdomen of the male with the first joint transverse, linear, twice the length of the second; the third much elongated, narrower towards its extremity, the apex nearly strait; the fourth subquadrate, with the anterior angles produced; fifth flask-shaped, with a very sharp extremity; of the female with the first five joints transverse, and of nearly an equal length; the sixth joint transverse quadrate, anteriorly notched, the last elongate, subtriangular behind, subproduced. Shell subcircular, the sides gradually converging into an angle behind; hinder part truncate and granulate-margined. Eyes narrower than their footstalks; orbits behind with two fissures, below with one.

Sp. 1. Heterodon. Shell granulated, the sides with seven serrulated teeth, and other smaller teeth between some of the other teeth; Front with three serrulated teeth, the middle of which is the largest.

Cancer (hippo) septem-dentatus.

Montagu, Trans. Linn. Soc. xi. tab. 1.

Atelecyclus septem-dentatus.

Leach, Edinb. Encycl. vii. 430.

Trans. Linn. Societ. xi. 313.

This elegant crab was discovered by Montagu on the southern coast of Devon, where it is not an uncommon species in deep water. To the fishermen it is well known by the name of old-man's face crab.

Division II. Subdivision 2.

Gen. 5. Portumnus. Leach.

Eyes not thicker than their peduncles; orbits entire. Anterior pair of legs equal; other legs with compressed claws, internally towards their base dilated; fifth pair with a compressed, foliaceous, lanceolate claw. Abdomen of the male with the fourth joint elongate. Shell with the transverse and longitudinal diameters the same.

Sp. 1. Variegatus. Shell obscurely granulated on each side with five teeth, the second and third somewhat obsolete; front with three teeth; wrists internally with one tooth.

Cancer (latipes) variegatus.

Planc, de Conch. Min. notis, p. 34. t. iii. f. 7.

B. C. male.

Cancer latipes.

Penn. Brit. Zool. iv. 3. t. 1. f. 4. female.

Portumnus variegatus.

Leach, Edinb. Encycl. vii. 391.

Malac. Podoph. Britann. t. iv. male and female.

Trans. Linn. Soc. xi. 314.

Planc first discovered this species on the shores of the Adriatic sea. It burrows beneath the sand, where it may be found by digging at low water, on most of our sandy shores.

When living it is most beautifully mottled, and the legs are sometimes of a luteous-orange colour.

Gen. 6. Carcinus. Leach.

Eyes narrower than their peduncles; orbits behind and beneath with one fissure. Anterior pair of legs unequal, the hands externally smooth; hinder pair compressed, and slightly formed for swimming. Abdomen of the male with the fourth joint transverse, and scarcely narrower than the third. Shell with the transverse diameter greatest.

Sp. 1. Menas. Shell with five teeth on each side; front with three rounded teeth or lobes; hands with one tooth, wrist with a spine.

Cancer Menas of authors. This most common species inhabits all the shores and estuaries of Britain. It burrows under the sand, or conceals itself beneath fuci and stones. It is sent to London in immense quantities, and is eaten by the poor.

Gen. 7. Portunus. Fabr. Latr. Bosc. Lam. Leach.

Eyes much thicker than their peduncles; orbits behind with two fissures, below with one fissure. Abdomen of the male with the fourth joint transverse. Anterior pair of legs somewhat unequal, the hands externally with elevated lines, arms generally unarmed; hinder pair compressed, foliaceous and formed for swimming. Shell with the transverse diameter greatest; the side with five, rarely with six teeth.

* Hinder claws with an elevated longitudinal line; external double palpi, with the second joint of their internal footstalk, truncate at the internal apex.

a. Orbits at the insertion of the antennae imperfect.

Wrists bidentate.

Sp. 1. Puber. Antennae half the length of the body, shell pubescent, front with many teeth.

Cancer puber. Linn. Syst. Nat.

Cancer velutinus. Penn. Brit. Zool. iv. 8. pl. iv. fig. 8.

Portunus puber.

Latr. Gen. Crust. et Ins. i. 27.

Leach, Edin. Encycl. vii. 390.

Trans. Linn. Soc. xi. 315.

Inhabits the rocky shores of the Mediterranean Sea and European Ocean. It is very common all along the southern coasts of Devon. In France it is used as an article of food.

b. Orbit internally slightly imperfect. Wrists unidentate.

Sp. 2. Corrugatus. Shell convex, with transverse serrate-granulated ciliated lines, the sides with five teeth on each side, the three hinder of which are more acute; front trilobate, the lobes subgranulate-serrate, the middle one largest; hands above unidentate, hinder claws with sharp points.

Cancer corrugatus.

Penn. Brit. Zool. iv. 5. pl. v. fig. 9.

Portunus corrugatus.

Leach, Edin. Encycl. vii. 390.

Trans. Linn. Soc. xi. 315.

Inhabits the British Seas. Pennant observed it opposite to Loch Jura in Sky, and the young state has been taken by C. Prideaux, Esq. in the Plymouth Sound.

** Hinder claws without the elevated line. External double palpi with the internal apex of the second joint of the internal footstalk emarginate. Orbits internally beneath the insertion of the antennae imperfect.

Sp. 3. Marmoreus. Shell convex, obsoletely and slightly granulated, with five nearly equal teeth on each side; front with three equal teeth, with rounded points; hands smooth, with one tooth above; hinder tarsi with acute points.

Cancer (pinnatus) marmoreus.

Montagu's MS.

Portunus marmoreus.

Leach, Edin. Encycl. vii. 390.

Malacost. Podophth. Britan. tab. 8.

Trans. Linn. Soc. xi. 317.

This elegant species, which derives its name from its colour, was discovered by G. Montagu, Esq. It is very common on the sandy shores of southern Devon, from Torcross to the mouth of the river Ex, and is frequently found entangled in the shore-nets of the fishermen, or thrown on the shore after storms.

It is distinguished from every other discovered species, by the rounded dentations of the front, the very slight elevation of the lines on the hands, and by the convexity, remarkable smoothness, and marbled appearance of the shell.

Young specimens are plain brown, and much resemble the young of P. depurator, from which they may very easily be separated by their more considerable convexity.

Gen. 8. Lupa. Leach.

Eyes much thicker than their peduncles; orbit above with two fissures, beneath with one fissure. Anterior pair of legs equal, the arms anteriorly spinose; hinder pair very much compressed. Abdomen of the male with the fourth joint much lengthened, much narrower than the third. Shell very transverse, furnished on each side with nine teeth, the last of which is the longest.

This genus was instituted by Dr Leach in the Edinburgh Encyclopaedia, and has since been given, with amended characters, in the Zoological Miscellany, and in the eleventh volume of the Transactions of the Linnean Society. As far as we have been enabled to learn, all the species inhabit the Great Ocean, harbouring amongst floating marine plants, swimming with great swiftness and ease near the surface of the water.

* Shell on each side terminated by a very long spine.

a. Fingers very long and filiform, hands externally smooth.

Sp. 1. Forceps. Shell granulated, wrists with a spine on each side, hands above at the base, and externally at the base, with one spine; fingers slightly bending upwards, and denticulated within; hinder claw very much compressed, round ovate.

Portanus forceps.

Fabr. Suppl. Ent. Syst. 368.

Bosc. Hist. Nat. des Crust. i. 220.

Cancer forceps. Herbst, iii. t. 55. fig. 4.

Lupa forceps.

Leach, Zool. Miscel. i. 123. t. 54.

Trans. Linn. Societ. xi. 319.

Plate XXI. This curious Lupa inhabits the Caribbean Sea. It was first figured by Dr P. Browne, in his History of Jamaica.

b. Fingers moderately long; hands externally with elevated lines.

Sp. 2. Trispinosa. Shell granulated, arms anteriorly with three spines.

Lupa trispinosa. Leach, Trans. Linn. Soc. xi. 319.

** Shell with the hinder lateral spine not very long.

Sp. 3. Banksii. Shell pubescent, front with four teeth, arms anteriorly with five teeth.

Lupa Banksii. Trans. Linn. Soc. xi. 319. Gen. 9. Matuta.

Dald. Fabr. Lam. Bosc, Latr. Leach.

External double palpi, with the internal footstalk elongate-subtriangular, the second joint with the internal side excavated and palpigerous. Fourth pair of legs, with the claw acute and narrower than the others. Shell very transverse on each side, terminated by a long spine.

Sp. 1. Victor. Shell on every side punctate, not striated behind.

Matuta victor. Latr. Gen. Crust. et Insect. i. 42.

Inhabits the Indian Ocean.

A. Division II. Subdivision 4.

Gen. 10. Cancer of authors.

External antennae short, inserted between the internal canthus of the eye and the front; internal antennae placed in foveole in the middle of the clypeus, with their peduncle nearly lunate. External double palpi with the second joint of the internal footstalk notched at the internal apex. Shell margined behind; orbits behind with one fissure, and externally with one fold; beneath with one fissure and externally with one fold. Anterior pair of legs unequal.

Sp. 1. Pagurus. Shell granulated with nine folds on each side; front with three lobes.

Cancer pagurus of authors.

This species is the common crab of Britain. It is considered to be in season between Christmas and Easter, and about harvest, being much esteemed as an article of food. Its natural history is but little known. During the summer months, it is very abundant on all our rocky coasts, especially where the water is deep. At low tide, they are often found in holes of rocks, in pairs, male and female; and if the male be taken away, another will be found in the hole at the next recess of the tide. By knowing this fact, an experienced fisherman may twice a day take, with little trouble, a vast number of specimens, after having once discovered their haunts.

In the winter, they are supposed to burrow in the sand, or to retire to the deeper parts of the ocean. They are taken in wicker-baskets, resembling mousetraps, or in large nets with open meshes, which are placed at the bottom of the ocean, and baited with garbage.

Gen. 11. Xantho. Leach.

External antennae very short, inserted in the internal corner of the eye; internal antennae received in foveole under the prominent margin of the clypeus, the peduncle sublinear. External double palpi, with the second joint of the internal footstalk notched at the internal apex. Shell submargined behind; orbits entire above, below externally with one fissure; anterior pair of legs unequal.

Sp. 1. Florida. Wrists above with two tubercles; shell on each side with four obtuse teeth; the interstices cut out; fingers black.

Cancer floridus.

Montagu, Trans. Linn. Soc. xi. 85. t. 2. f. 1.

Cancer incisus. Leach, Edin. Encycl. vii. 391.

Xantho incisa. Leach, Edin. Encycl. vii. 430.

Xantho florida. Leach, Trans. Linn. Soc. xi. 320.

This species was first described by Montagu, who Crustacea considered it to be the Cancer floridus of Linné; but an examination of the characters in the Amoenitates Academicae, will readily convince the naturalist of the incorrectness of this opinion; nor is it the Cancer floridus of Herbst; which induces me to believe that some one must have misled Mr Montagu with regard to the synonym, as he could never have considered them the same, had he examined the reference.

Cancer donone of Herbst, probably is referable to the genus Xantho.

Gen. 12. Calappa. Dald. Fabr. Lam. Latr. Bosc, Leach.

Shell with the hinder angles arched, receiving the legs when contracted. Hands crested, and equal.

One of the species is very common in the Mediterranean Sea, and is mentioned by Aristotle and Athenæus. Of the economy of the genus nothing is known, but, from the general structure of the parts, it is probably similar to that of Cancer.

Sp. 1. Granulatus. Shell tuberculated, the hinder angles dentated; hinder teeth strong, acute; hinder margin subemarginated at the base of the tail.

Calappa granulata.

Fabr. Supl. Ent. Syst. 346.

Latr. Gen. Crust. et Insect. i. 28.

Bosc, Hist. Nat. des Crust. i. 184.

Leach, Edinb. Encycl. vii. 391.

Inhabits the Mediterranean Sea.

B. Division III. Subdivision 1.

Gen. 13. Pilumnus. Leach.

External double palpi, with the second joint of the internal footstalk with the internal apex truncate-emarginate. Claws simple, with naked tips.

Sp. 1. Hirtellus. Body and legs bristly; shell with five teeth on each side; claws somewhat muricated on the outside.

Cancer hirtellus.

Linn. Syst. Nat. 1045.

Penn, Brit. Zool. iv. 6.p. 6. f. n.

Leach, Edinb. Encycl. vii. 391.

Inhabits the south coast of Devonshire.

Gen. 14. Gecarcinus. Leach.

External double palpi, with the internal footstalk composed of two nearly equal joints; palpi inserted beneath or within; anterior pair of legs, unequal; claws and tibiae spinose; shell truncate-subcordate.

There are probably many species of this genus, but their characters have not been made out by naturalists. Travellers speak of three sorts. Their economy is very curious, and is detailed by Sloane, Catesby, and others.

They inhabit the mountains of the Antilles. In the beginning of the year they descend from the mountains in vast numbers, and deposit their eggs in the sea. During this march, which is said to be conducted with great regularity, and to be under the guidance of a commander, the strongest proceed first. They are often obliged to halt for want of rain, and to go into the most convenient encampment until the weather change. The main body is said to consist of females, which never leaves the mountains till the rain be set in. They chiefly proceed during the Crustacea, night, sheltering in hollow trees and in shady places during the hot part of the day.

Sp. 1. *Ruricola*. Tarsi with six elevated, serrated lines; hands smooth.

Cancer ruricola.

Linnaeus, *Syst. Nat.* 2040. Fabre, *Suppl. Ent. Syst.* 389. Herbst, tab. 3, f. 36.

Ocypode ruricola.

Bosc, *Hist. Nat. des Crust.* i. 196. Leach, *Edinb. Encycl.* vii. 393.

Gecarcinus ruricola.

Leach, *Edinb. Encycl.* vii. 330. — *Trans. Linn. Soc.* xi. 322.

Inhabits Southern America.

B. Division III. Subdivision 2.*

Gen. 15. *Pinnotheres*. Latreille, Bosc, Leach.

Alphaeus. Daldorff.

Antennae very short (the three first joints largest), inserted in the interior corner of the eyes. External double palpi, with the internal footstalk, one-jointed. Anterior pair of legs unequal. Eyes thick. Shell ovate-orbicular, orbiculate-quadrate, or transverse-subquadrate.

All the species of this most interesting genus inhabit the bivalve shells of the acephalous Mollusca, and were supposed by the ancients to be consanguineous inmates with the animal, bound by mutual interest. The fable is beautifully told by Oppian, and is alluded to by Cicero. "Pinna vero (sic enim Graece dictatur), duabus grandis patulis conchis, cum parva squilla quasi societatem coit comparandi cibi. Itaque cum pisciculi parvi in concham hiantem innotaverint, tum admonita a squilla pinna morsu, comprimit concham." *Nat. Deor. Lib.* ii. sec. 48.

Aristotle supposed them to act as sentinels, and believed that they guarded the pinna (the animal in whose shell they were first observed), from the attacks of its enemies. Rondeletius and some other naturalists held the same opinion.

Sp. 1. *Cranchii*. Shell orbiculate-subquadrate, soft, very smooth, with the sides dilated behind; front strait, obscurely submarginate; hands oblong, below and the thighs above with a ciliated line; thumb subcarinate; abdomen very broad, the sides of the segments arcuate, the second, and following ones distinctly notched; the fifth segment somewhat broader; the last narrower than the preceding segment. Female.

Pinnotheres Cranchii.

Leach, *Malacost. Podophth. Britann.* tab. 14, fig. 4-5.

The male of this species, which was discovered by Mr. J. Cranch, whose name it bears, is unknown. It is distinguished from *P. pisum* (the common species), by the form of the front of the shell, which is strait and slightly notched; by the dilated hinder part of the shell, and by the abdomen, all the joints of which, excepting the first, are distinctly notched behind.

Plate XXI.

Division III. Subdivision 2.**

Gen. 16. Ocypode.

Daldorff, Fabre, Latreille, Lamarck, Bosc, Leach.

Eyes very large, with their peduncle generally extending beyond their tips, in the form of a spine. Anterior pair of legs unequal.

The economy of the Ocypodes is unknown, but it is probably not very different from that of the Uca. They inhabit the shores of the sea; and, as their name indicates, run with great swiftness.

Sp. 1. *Ceratophthalma*. Arms granulated; hands cordate; spine of the peduncle of the eye smooth and very long.

Ocypode Ceratophthalma.

Fabre, *Suppl. Entom. Syst.* 465. Latreille, *Gen. Crust. et Ins.* i. 32. Leach, *Trans. Linn. Soc.* xi. 322.

Inhabits the shores of the Indian Ocean.

Gen. 17. Uca. Leach.

Ocypode. Latreille, Bosc, Fabre &c.

Eyes small, terminating their peduncle. Anterior pair of legs very unequal.

The Uca inhabit the shores of the sea, swamps, and the banks of rivers, living in holes beneath the ground. They multiply very much, and serve as food to otters, bears, birds, crocodiles, &c. The females carry about their eggs under the abdomen. They live on animal substances, which they devour with avidity. Their motion is rapid; Sir J. Banks assures us that he could never come up with them when running, and that the only means by which he could take them was by running to and stopping their holes before they could reach it, and which he effected by lying in ambush until the animals had gone to a very considerable distance from their haunts.

Sp. 1. *Una*. Shell with one tooth on each side.

Cancer vocans major. Herbst, t. i. f. 10.

Cancer vocans. Bosc, *Hist. Nat. des Crust.* i. 198.

Uca una. Leach, *Trans. Linn. Soc.* xi. 323.

Inhabits South America.

Gen. 18. *Gonoplax*. Leach.

Ocypoda. Bosc.

Eyes terminating their peduncle. Anterior pair of legs equal; of the male very long; of the female twice the length of the body. Antennae half of the length of the body, inserted at the internal canthus of the eyes.

The animals of this genus inhabit the ocean, preferring such parts as have a slimy bottom. They burrow laterally in the clay or slime, making two entrances to their hole; entering by one, and going out by the other. We have seen several fossil species inclosed in marle, that appear to belong to this genus.

Sp. 1. *Bispinosa*. Shell on each side with two spines, arms above, and wrists internally with one spine.

Cancer angulatus. Pennant, *Brit. Zool.* iv. t. 5, f. 10. Fabre, *Suppl. Entom. Syst.* 341.

Ocypoda angulata. Bosc, *Hist. Nat. des Crust.* i. 198.

Gonoplax bispinosa.

Leach, *Edinb. Encycl.* vii. 430. — *Trans. Linn. Soc.* xi. 323.

Inhabits the British Sea. It is not uncommon at Salcombe and in the Plymouth Sound; likewise occurs at Weymouth, and at Redwharf in Anglesea.

Division III. Subdivision 3.

Gen. 19. *Grapsus*. Lamarck, Latreille, Bosc, Leach. Sp. 1. *Pictus*. Shell with the sides plicated behind, anteriorly with the angles bidentate; front with four dentated folds; fingers with their joints concave; wrists internally strongly unidentate.

*Cancer grapsus*.

*Linn. Syst. Nat.* 1048. *Fabr. Supl. Ent. Syst.* 342.

*Grapsus pictus*.

*Latr. Gen. Crust. et Insect.* i. 33. *Leach, Edinb. Encycl.* vii. 393. *, Trans. Linn. Soc.* xi. 325.

Inhabits the South American Ocean.

B. Division IV. Subdivision 1.

Gen. 20. *Homola*. *Leach*.

Shell elongate-quadrate, a little produced in front; eyes large, somewhat globose, their footstalks lengthened, the second joint very short and very abruptly thicker than the first. *External antennae* very long, inserted beneath the eyes, the two first joints long, the first thickest; *internal antennae* inserted within the orbit of the eye, and capable of being lodged in the internal corner. *External double palpi* with their internal footstalk composed of two lengthened and narrow joints; *palpi* three-jointed; the first joint shortest. *Legs* ten; first pair largest and didactyle; the three following pair simple, alike in form, and having their *claws* spiny within; fifth pair monodactyle, the *claws* and *tarsi* being spiny within. *Abdomen* composed of seven joints.

Sp. 1. *Spinifrons*. Shell anteriorly spinous; sides anteriorly beset with small spines; hinder thighs internally with three spines.

*Homola spinifrons*.

*Leach, Trans. Linn. Soc.* xi. 324. *, Zoolog. Miscel.* ii. 82. t. 88.

Inhabits the Mediterranean Sea.

B. Division IV. Subdivision 2.

Gen. 21. *Dorippe*. *Dald.* *Fabr. Lam.* *Latr. Bosc.*, *Leach*.

*External double palpi* with the internal footstalk having its first joint dilated internally, the second narrow, bearing palpi at its extremity. *Shell* subtriangular, anteriorly truncated. Second and third pair of legs alike, with acute, elongate, simple subquadrate *claws*; fourth and fifth pair shorter, dorsal and monodactyle. *External antennae* inserted above and between the eyes; *internal antennae* inserted below and between the eyes.

Sp. 1. *Quadridenta*. Middle of the clypeus with four teeth, the lateral ones shortest; sides of the shell unidentate at their middle part; four anterior thighs subdentate.

*Dorippe quadridenta*.

*Fabr. Supl. Ent. Syst.* 361. *Bosc, Hist. Nat. des Crust.* i. 207. *Latr. Gen. Crust. et Insect.* i. 41. *Leach, Edin. Encycl.* vii. 395. *, Trans. Linn. Societ.* xi. 324.

Inhabits the Mediterranean Sea.

Gen. 22. *Dromia*. *Dald.* *Fabr. Latr. Bosc*, *Leach*.

*External double palpi* two-jointed, with the second joint somewhat broader, shorter, and at its interior apex palpigerous. Second and third pair of legs simple; fourth and fifth shorter and didactyle. *External antennae* inserted below the eyes; the two first joints largest, and abruptly thicker than the others; *internal antennae* inserted below, and somewhat between the eyes.

Sp. 1. *Rumphii*. Shell hairy, on each side with five strong teeth, without any remarkable interval between them; arms and legs without knots.

*Cancer Dromia*. *Linn. Syst. Nat.* 1043. *Dromia Rumphii*.

*Fabr. Supl. Ent. Syst.* 359. *Leach, Edin. Encycl.* vii. 391.

Inhabits the East Indian Ocean. It is named after Rumphius, by whom it was first described and figured. *Rarit. Amb.* t. 11. No. 1.

B. Division V. Subdivision 1.

Gen 23. *Eurynome*. *Leach*.

*External antennae* rather long, with the first joint shorter than the second. *Shell* verrucated, anteriorly terminated by a bifid rostrum, with divaricating laciniæ. Eyes distant, thicker than their peduncle, which is of moderate length. *External double palpi* with the interior point of the second joint of their internal footstalks truncate-emarginate. *Anterior legs* equal; of the male three times the length of the body; of the female longer than the body.

Sp. 1. *Aspera*. Anterior legs and thighs tuberculated; shell with eight tubercles on the back that are more elevated than the others, which are irregular and margined with hairs; the sides with four lamellæ; rostrum with simple acuminate laciniæ.

*Cancer asper*. *Penn. Brit. Zool.* iv. 8.

*Eurynome aspera*.

*Leach, Edin. Encycl.* vii. 481. *Malac. Podophth. Britan*. *, Trans. Linn. Societ.* xi. 326.

Inhabits the British Sea.

Gen. 24. *Parthenope*. *Fabricius*, *Leach*.

*Maja I*. *Latreille*.

*External antennae* very short; the two first joints largest, the first more so than the second. *Shell* tuberose, anteriorly acuminated; the rostrum entire; eyes distant, not thicker than their peduncles, which are very short. *External double palpi* with the internal apex of the second joint of their internal footstalk truncate-emarginate. *Anterior legs* unequal; of the male very thick.

Sp. 1. *Horrida*. Tubercles of the shell impressed; the punctures as if eaten; legs spiny; hands and wrists verrucated; abdomen and breast carious.

*Cancer longimanus spinosus*.

*Seb. Mus.* iii. 48. tab. 19. fig. 16, 71. *Cancer horridus*. *Linn. Syst. Nat.* i. 104. *Parthenope horrida*.

*Fabr. Suppl. Ent. Syst.* 353. *Leach, Edin. Encycl.* vii. 431. *, Zoolog. Miscel.* ii. tab. 93.

*Maja horrida*.

*Latr. Gen. Crust. et Insect.* i. 37. *Leach, Edin. Encycl.* vii. 394.

Inhabits the Asiatic Ocean.

B. Division V. Subdivision 2.

Gen. 25. *Pisa*. *Leach*. Blastus. Leach.

External antennae with clubbed hairs, the first joint longer than the second. External double palpi with the second joint of the internal footstalk with its internal apex truncate or emarginate. Claws internally denticulated. Shell villose, the laciniæ of the rostrum divaricating. Orbits behind with two, below with one fissure.

* Shell densely villose, the sides on each side behind terminated with a spine.

Sp. 1. Gibbii. Rostrum descending; shell with a spine behind the eyes on each side; arms and thighs simple.

Cancer biauculatus.

Moulay, Trans. Linn. Soc. xi. 2. t. 1. f. 1. Pisa biauculata. Leach, Edin. Encycl. vii. 431. Pisa Gibbii.

Leach, Trans. Linn. Soc. xi. 327. — Molac. Podophth. Britan. tab. 19.

Inhabits deep water on the coasts of Devon and Cornwall.

Sp. 2. Nodipes. Rostrum horizontal; arms and tips of the thighs knotted.

Pisa nodipes.

Leach, Zool. Miscell. ii. t. 78. — Trans. Linn. Soc. xi. 328.

** Shell villose, with spiny sides.

Sp. 3. Tetragonon. Shell on each side with six spines; two small, the rest larger.

Cancer tetraodon. Penn. Brit. Zool. iv. 7. t. 8. f. 15. Maja tetraodon. Bosc, Hist. Nat. des Crust. i. 254. Blastus tetraodon. Leach, Edin. Encycl. vii. 431. Pisa tetraodon. Leach, Trans. Linn. Soc. xi. 328.

Inhabits the south-west coast of England.

Gen. 26. Lissa. Leach.

Maja. Latr. Bosc.

External antennae with clubbed hairs, the first joint longer than the second. External double palpi with the second joint of the internal footstalk truncate-emarginate at the internal apex. Claws internally simple, with naked points. Shell tuberose, the laciniæ of the rostrum meeting together. Orbits with one fissure behind, and one below.

Sp. 1. Chiragra. Rostrum obtuse, with the anterior angles subreflexed; legs knotted.

Cancer chiragra. Herbst, tab. 17. f. 96. Inachus chiragra. Fabr. Supl. Ent. Syst. 357. Maja chiragra.

Bosc, Hist. Nat. des Crust. i. 255. Latr. Hist. Nat. des Crust. et des Insect. vi. 97. Lissa chiragra. Leach, Zool. Miscel. ii. 70. t. 73.

Inhabits the Mediterranean Sea.

Division V. Subdivision 3.

Gen. 27 Maja. Lam. Latr. Bosc, Leach.

External antennae with two first joints thickest, and of nearly equal length. Shell convex ovate-subtriangular, very spiny. Eyes not thicker than their elongate peduncle. External double palpi with the second joint of their internal footstalk, deeply notched at its internal apex. Claws with naked sharp points.

Sp. 1. Squinado. Shell fasciculate-pilose, orbit above with one spine, the sides with five strong spines, clypeus beneath the front with a short spine excavated.

Cancer squinado.

Herbst, iii. t. 56. full grown. — i. t. 14. f. 85—84. junior.

Cancer maja.

Scopoli, Entom. Carn. 1126. Sowerby, Brit. Miscell. t. 39.

Maja squinado.

Latr. Gen. Crust. et Insect. i. 37. Bosc, Hist. Nat. des Crust. i. 257. Leach, Edinb. Encycl. vii. 394—431. — Trans. Linn. Soc. xi. 326.

Inhabits the southern coasts of Devon and Cornwall. By the fishermen it is named Thornback, or king-crab.

B. Division V. Subdivision 4.

Gen. 28. Egeria. Leach.

Shell spinous anteriorly, terminated by an elongate rostrum; eyes large, much thicker than their peduncles; orbits behind with two fissures, below with one fissure. External antennae inserted at the sides of the rostrum with the two first joints thickest, the second joint much thicker than the first. Two anterior legs of the male about twice the length of the body, filiform, and scarcely thicker than the others; eight hinder legs very long very slender, alike in form, in order of size 2, 3, 4, and 5. Claws elongate, somewhat arcuate, and very slender. External double palpi with the second joint of the internal footstalk with its internal side strait, the interior apex being abruptly produced.

Sp. 1. Indica. Beak acutely notched; shell behind the beak with six tubercles, arranged in transverse lines, 3, 2, 1, 1.

Doclea Indica. Leach, Zool. Miscel. ii. 40. tab. 73.

Inhabits the Indian Ocean, where it is by no means uncommon.

Gen. 29. Doclea. Leach.

Shell villose, with the sides somewhat spinous; front terminated by a short beak; eyes moderate in size, but much thicker than their peduncles; orbits behind and below with one fissure. External antennae inserted at the base of the beak, the second joint shorter than the first. Anterior pair of legs of the female as long as the body, and more slender than the others, which are very long and slender; claws elongate, arcuate, slender. External double palpi with their internal footstalk having the internal side of the second joint towards the apex deeply notched.

Sp. 1. Rissonii. Shell and legs with brown pubescence; hinder part of the shell with one spine; each side with three spines.

Doclea Rissonii. Leach, Zool. Miscel. ii. 42. tab. 74.

Locality unknown.

B. Division V. Subdivision 5.

Gen. 30. Hyas. Leach.

Shell elongate-subtriangular, subtuberculated, the sides behind the eyes produced into a lanceolate projection; rostrum fissured, the laciniæ approximating. External antennae with the first joint dilated, larger than the second. External double palpi with the second joint emarginate at the internal apex. Sp. 1. *Araneus*. The lastiform process behind the eyes, tuberculated behind.

*Cancer araneus*.

*Linn. Syst. Nat.* 1044.

*Cancer bufo*.

*Herbst*, i. 142. t. 17. f. 59.

*Inachus araneus*.

*Fabr. Suppl. Ent. Syst.* 359.

*Hyas araneus*.

*Leach*, *Edin. Encycl.* vii. 437.

*Trans. Linn. Soc.* xi. 329.

Inhabits the Scottish Sea in great plenty; on the English coasts it is more rare.

**Division VI.**

Gen. 31. *Lithodes*. *Latreille*, *Leach*.

*External double palpi* with narrow subcylindric footstalks. *Eyes* approximating at their base. *Shell* very spiny, anteriorly rostrated.

Sp. 1. *Maja*. Legs and shell with sharp spines; beak spiny, with the tip bifurcate; fingers with tufts of hair.

*Cancer maja*.

*Linn. Syst. Nat.* 1046.

*Cancer horridus*.

*Penn. Brit. Zool.* iv. 7. pl. 7. fig. 14.

*Inachus maja*. *Fabr. Ent. Syst. Suppl.* 358.

*Maja vulgaris*.

*Bosc*, *Hist. Nat. des Crust.* i. 251.

*Lithodes arctica*.

*Latr. Gen. Crust. et Insect.* i. 40.

*Lithodes maja*.

*Leach*, *Edin. Encycl.* vii. 395.

*Trans. Linn. Soc.* xi. 332.

Inhabits the Northern Sea, and in our seas is very rare, or at least very local; occurring only on the rocky shores of Yorkshire, and of Scotland.

**C. Division VII. Subdivision 1.**

Gen. 31. *Inachus*. *Fabricius*, *Leach*.

Shell slightly spined, with a spine on each side protecting the eye when retracted. *Eyes* distant, scarcely thicker than their peduncles. *External double palpi*, with the second joint of the internal footstalk, truncate at its internal point. *External antennae* with the three first joints thickest. *Second pair of legs* thicker than the following ones. *Claws* curved.

Sp. 1. *Dorsettensis*. Beak short, emarginate, the clypeus beneath produced into a spine; shell anteriorly, with four little tubercles placed transversely; then with three spines (the anterior one strongest); behind with three strong sharp spines (the middle one generally longest and strongest), forming a slightly recurved line; hinder margin with two distinct obsolete tubercles.

*Cancer Dorsettensis*.

*Penn. Brit. Zool.* iv. 8. pl. 9. fig. 18.

*Cancer scorpio*.

*Fabr. Sp. Ins.* i. 504.

*Gmel. Syst. Natur.* i. 2078.

*Herbst*, i. 237, 130.

*Inachus scorpio*.

*Fabr. Ent. Syst. Suppl.* 358.

*Macropus scorpio*.

*Latr. Hist. Nat. des Crust. et des Insect.* vi. 109.

*Maja scorpio*.

*Bosc*, *Hist. Nat. des Crust.* i. 252.

*Inachus Dorsettensis*.

*Leach*, *Edin. Encycl.* vii. 431.

— *Malac. Podolph. Britton*. tab. 22. fig. 1—6.

— *Trans. Linn. Soc.* xi. 330.

Inhabits the British Seas.

**Division VI. Subdivision 2.**

Gen. 32. *Macropodia*. *Leach*.

*Macropus*. *Latreille*.

*Shell* slightly spined, beak long and fissured. *Eyes* distant, subreniform, much thicker than their peduncles. *External antennae* half the length of the body, the second joint three times the length of the first. *External double palpi* slender, the internal footstalk with two equal joints; *palpi* very hairy, the middle joint shortest, the third a little longer than the first. *Four anterior claws* with their tips bent; four hinder ones abruptly curved at their base.

Sp. 1. *Phalangium*. Beak acuminate, much shorter than the antennae; shell behind the rostrum with three tubercles placed in a triangle, the hinder tubercle largest; arms internally subscabrous and hirsute.

*Cancer phalangium*.

*Penn. Brit. Zool.* iv. 8. pl. 9. fig. 17.

*Macropus longirostris*.

*Latr. Gen. Crust. et Insect*.

*Macropodia longirostris*.

*Leach*, *Edin. Encycl.* vii. 395.

*Leach*, *Zool. Miscel.* ii. 18.

— *Trans. Linn. Soc.* xi. 331.

— *Malac. Podolph. Britton*. tab. 23.

Inhabits the mouths of rivers, and is very common in Great Britain.

**D. Division VIII.**

Gen. 33. *Leptopodia*. *Leach*.

*Shell* not spinous, the beak very long and entire. *Eyes* distant, globose. *External antennae* half the length of the body, the second joint three times the length of the third. *External double palpi* slender, the internal footstalk, with the second joint half the length of the first. *Palpi* very hairy, the last joint largest, the two first joints nearly equal. *Claws* long, alike in form, and slightly bent.

Sp. 1. *Sagittaria*. Hands granulose; beak on each side, and the arms and thighs anteriorly spinous.

*Inachus sagittarius*.

*Fabr. Suppl. Ent. Syst.* 359.

*Cancer sagittarius*. *Herbst*, 3.

*Macropus sagittarius*.

*Latr. Hist. Nat. des Crust. et des Insect.* vi. 112.

*Maja sagittaria*.

*Latr. Gen. Crust. et Insect.* i. 38, 4.

*Leach*, *Edin. Encycl.* vii. 395.

*Maja sagittis*.

*Bosc*, *Hist. Nat. des Crust.* i. 253.

*Leptopodia sagittaria*.

*Leach*, *Zool. Miscel.* ii. 16. tab. 67.

— *Trans. Linn. Soc.* xi. 332.

Inhabits the Caribbean Sea. Gen. 35. PACTOLUS. Leach.

Shell not spiny. Beak very long and entire. Legs of moderate length; the first, second (and third) pairs furnished with a simple claw; the fourth and fifth pairs didactyle.

The abdomen of the female has the first joint narrow, the second, third, and fourth joints transverse-linear, the fifth very large and somewhat rounded, as in the case with the genus Leptopodia.

Sp. 1. Bozzi. Beak on each side sinuose; legs ciliate-punctate.

Pactolus Bozzi.

Leach, Zool. Miscel. ii. 20. tab. 68.

Trans. Linn. Soc. xi. 333.

A single specimen of this curious animal is preserved in the British Museum; but its locality is not known. Fabricius seems to have described it as the other sex of his Inachus sagittarius.

E. Division IX.

Gen. 36. LEUCOSIA. Fabr. Latr. Bosc, Leach.

Shell rounded or rhomboidal, slightly produced in front. External double palpi with the second joint of their internal footstalk simple. Anterior pair of legs distinctly thicker than the others, which are simple.

This genus requires to be investigated. It contains two indigenous species; namely Cancer tumefactus of Montagu, and Cancer tuberosus of Pennant.

* Second joint of the internal footstalk of the external double palpi dilated.

Sp. 1. Anatum.

Cancer anatum.

Herbst, i. 95. tab. 3. fig. 19.

** Second joint of the internal footstalk of the external double palpi nearly linear.

Sp. 2. Craniolaris.

Cancer craniolaris. Herbst, i. 90. tab. 2. fig. 17.

Gen. 37. Ixa. Leach.

Shell very transverse, subcylindric, much broader than long. External double palpi with the second joint of the internal footstalk excavated. Anterior pair of legs scarcely thicker than the rest.

Sp. 1. Cylindrus. Shell with two channels, the sides rough and terminated by a spine.

Leucosia cylindrus.

Latr. Hist. Nat. des Crust. et des Insect. vi. 119.

Cancer cylindrus.

Herbst, i. 108. tab. 2. f. 29, 30, 31. male.

Ixa cylindrus.

Leach, Trans. Linn. Soc. xi. 334.

Inhabits the Indian Ocean.

Brachyurous Genera of uncertain situation.

Gen. 38. HEFATUS. Latreille.

External double palpi with the second joint of their internal footstalk elongate-triangular, gradually becoming sharp from the base to the point. Shell arcuate before, the sides converging behind. Legs all formed for walking; anterior pair didactyle, the hands crested.

Sp. 1. Fasciatus. Latreille.

This species is figured by Herbst, tab. 38. fig. 2. The shell and legs are banded with red, sometimes with brown.

Gen. 39. PLAGUSIA. Latreille.

VOL. I. PART II.

Eyes with a short peduncle, inserted at the anterior angles of the shell. Shell quadrate. Interior antennae inserted in two fissures on the clypeus.

Sp. 1. Depressa. Middle of the clypeus with two teeth; sides of the shell with five teeth; dorsal tubercles naked.

Cancer depressus. Fabr. Ent. Syst. Suppl. 343.

Plagusia depressa. Latr. Gen. Crust. et Insect. i. 34.

Inhabits the shores of the Mediterranean Sea.

Gen. 40. MICTYRIS. Latreille.

External double palpi with the first joint very large. Antennae very short. Shell subovate, truncate behind, elevated. Arms at the base of the wrist bent like a knee.

Sp. 1. Longicarpus.

Latr. Gen. Crust. et Insect. i. 41.

Leach, Edinb. Encycl. vii. 395.

Gen. 41. ORITHYIA. Daldorff, Fabricius, Latreille, Boac. Leach.

Shell rounded. Legs all placed in the same horizontal line, the hinder pair with the last joint compressed, or formed for swimming; the first pair didactyle.

Sp. 1. Mamillaris.

Orithya mamillaris.

Fabr. Suppl. Ent. Syst. 363.

Latr. Gen. Crust. et Insect.

Inhabits the Indian Ocean. It is figured by Herbst, tab. 18. f. 101.

Gen. 42. RANINA. Lamarck, Latreille, Leach.

Legs, with the exception of the first pair, which is monodactyle, formed for swimming; the two hinder pairs placed above the others.

Sp. 1. Serrata. Arms much dentated; front of the shell with dentated lobes.

Ranina serrata.

Latr. Gen. Crust. et Insect. i. 43.

Leach, Edinb. Encycl. vii. 396.

Inhabits the Indian Ocean.

Gen. 43. MEGALOPA. Leach.

This genus contains but one species, which is described in the seventh volume of the Transactions of the Linnean Society, under the title Cancer rhomboidalis. Cancer granarius of Herbst probably belongs to the same genus. See Trans. Linn. Soc. vii. tab. vi. fig. 1.

Order II. MACROURA.

This order contains the families, Pagurii, Palinurini, Astacini, and Squillares of Latreille.

Synopsis and distribution of the Genera.

A. Tail on each side with simple appendages.

Division I. Legs ten; anterior pair largest and dactyle.

Genus 44. PAGURUS.

45. Birgus.

B. Tail on each side with foliaceous appendages, forming with the middle tail-process a fan-like fin.

a. Interior antennae with very long footstalks.

Division II. External antennae squamiform. Legs ten, alike and simple.

Genus 46. SCYLLARUS.

47. THENUS.

Division III. External antennae setaceous, and very long. Legs ten, alike and simple. Genus 48. Palinurus.

Division IV. External antennae very long and setaceous. Legs ten, anterior pair didactyle, fifth pair spurious.

Genus 49. Porcellana.

50. Galatea.

b. Interior antennae with moderate peduncles.

Division V. Exterior lamella of the tail simple. Antennae inserted in the same horizontal line, the interior ones with two setae, the exterior ones simple. Legs ten.

Genus 51. Gebia.

52. Callianassa.

53. Axis.

Division VI. Exterior lamella of the tail bipartite. Antennae inserted in the same horizontal line, the internal ones with two setae, the external ones with a spine-shaped squama at the first joint of the peduncle. Legs ten (anterior pair largest and didactyle).

Genus 54. Astacus.

55. Nephrops.

Division VII. External antennae with a large broad squama or scale at their base. Abdomen with the second joint anteriorly and posteriorly produced below. Legs ten.

Subdivision 1. External antennae inserted in the same horizontal line with the interior ones, which have two setae. Tail with the external lamella composed of but one part.

Genus 56. Crangon.

Subdivision 2. External antennae inserted below the internal ones; interior ones with two setae inserted in the same horizontal line. Exterior lamella of the tail bipartite.

Genus 57. Atya.

58. Processa.

Subdivision 3. External antennae inserted below the internal ones; interior ones with two setae, one placed above the other. (External lamella of the tail composed of but one part).

• Internal antennae with the superior seta excavated below. Claws spinulose.

Genus 59. Pandalus.

60. Hippolyte.

61. Alpheus.

** Internal antennae with the superior seta not excavated. Claws simple.

Genus 62. Penaeus.

Subdivision 4. External antennae inserted below the internal; interior ones with three setae. (External lamella of the tail composed of but one part).

Genus 63. Palemon.

64. Athanas.

Division VIII. External antennae inserted below the internal ones, with a large scale at their base. Legs sixteen.

Genus 65. Mysis.

C. Tail with two setae, one on each side.

Division IX.

Genus 66. Nebalia.

The genera whose situation has not been ascertained are the following, namely,

Genus 67. Albunea.

68. Remipes.

69. Hippa.

Genus 70. Thalassina.

71. Squilla.

Division I.

Gen. 44. Pagurus. Fabr. Latr., Bosc, Leech, &c.

External antennae, with the second joint of their peduncle, with a moveable spine affixed to the apex above. Abdomen membranaceous. Tail three-jointed, crustaceous, the second joint on each side appendiculated. Four hinder legs spurious, short, didactyle.

The curious economy of the genus Pagurus attracted the attention of the ancients. One species is well described by Aristotle under the name nymphæ.

All the species are parasitical, and inhabit the cavities of turbinated univalves. They all change their habitation during their growth, first occupying the smallest shells, and latterly those of very considerable dimensions. The abdomen is naked and slender, being covered merely with a skin of a delicate texture; but its extremity is furnished with appendages, by means of which it secures itself within the shell of which it makes choice. It is really astonishing with what facility these animals move, bearing at the same time the shell, which is destined to preserve the body from injury, and to guard these animals from the attacks of fishes, who would otherwise devour them. All the species are termed indiscriminately Soldier-crabs and Hermit-crabs, from the idea of their living in a tent, or retiring to a cell.

Sp. 1. Bernhardus (common soldier-crab). Arms hairy, nuicated, the left the largest; hands subordinate, fingers broad.

Pagurus, Bernhardus.

Fabr. Suppl. Ent. Syst. 411.

Latr. Gen. Crust. et Insect. i. 46.

Leach, Edinb. Encycl. vii. 396.

Cancer Bernardus. Linn. Syst. Nat.

Inhabits the European Ocean, and is very abundant in the British Seas, inhabiting various kinds of univalve shells, changing its habitation as it grows. Pagurus araneiformis, Edinb. Encycl. vii. 396, is merely the young of this species.

Gen. 45. Birgus. Leach.

External antennæ with the second joint of its peduncle crested. Abdomen crustaceous. Tail two-jointed, crustaceous, the first joint on each side appendiculated. Fourth pair of legs didactyle; fifth pair (didactyle?).

Sp. 1. Latro. Shell anteriorly with a simple acuminate rostrum.

Cancer latro. Linn. Syst. Nat. 1049.

Cancer (astacus) latro. Herbst, ii. 34, tab. 24.

Pagurus latro.

Fabr. Ent. Syst. ii. 468.

Leach, Edinb. Encycl. vii. 390.

Birgus latro, Leach, Trans. Linn. Soc. xi. 337.

This species is said to inhabit Amboyna, and to live in cavities and holes of rocks, from whence it wanders abroad in the night, in order to procure cocoanuts, on which it is supposed to feed.

Division II.

Gen. 46. Scyllarus.

Fabr. Dalld. Lam. Latr. Bosc, Leach.

Hinder legs with the tarsi beneath produced into **ANNULOSA.**

*Crustacea.* a thumb. *Thorax* convex, sublinear. *Eyes* inserted behind the exterior antennae.

Sp. 1. *Arcus.* External antennae very much dentated; shell above with a triple series of dentations. Cancer arcus. *Linn. Syst. Nat.* 1053.

Seyllarus arcus.

*Latr. Gen. Crust. et Insect.* i. 47.

*Leach, Edinb. Encycl.* vii. 397.

Inhabits the European Ocean, and is said by Pennant to have been taken in the British Sea.

Gen. 47. *Thenus.* Leach.

*Hinder legs* with simple tarsi. *Thorax* subdepressed, broader anteriorly. *Eyes* inserted at the anterior angles of the thorax.

Sp. 1. *Indicus.* External antennae serrated; thorax granulated, carinated, trispinous; abdomen granulated, the granules arranged transversely.

Inhabits the Indian Ocean.

**Division III.**

Gen. 48. *Palinurus.*

*Dald. Fabr. Lam. Latr. Bosc, Leach.*

The animals of this genus have the power of producing a sound by rubbing their exterior antennae against the sides of the projecting clypeus.

Sp. 1. *Vulgaris.*

Astacus homarus. *Penn. Brit. Zool.* iv. 16. pl. 11.

Inhabits the European Ocean. It is commonly eaten in London, and is sometimes denominated spiny-lobster or sea cray-fish.

**Division IV.**

Gen. 49. *Porcellana.* Lam. Latr. Bosc, Leach.

*External double palpi* with the first joint of the internal footstalk dilated internally. Shell orbiculate subquadrate.

Sp. 1. *Platycheles.* Anterior margin of the shell with three entire teeth; claws very large and much depressed; wrists internally denticulated; hands externally deeply ciliated.

Cancer platychelis.

*Penn. Brit. Zool.* iv. 6. pl. 6. and 12.

Porcellana platychelis.

*Latr. Gen. Crust. et Insect.* i. 49.

*Leach, Edinb. Encycl.* vii. 398.

*Trans. Linn. Soc.* xi. 339.

Inhabits the rocky shores of the southern and western coasts of Britain, concealing itself beneath stones, to the under side of which it adheres closely.

Gen. 50. *Galatea.*

*Galathea.* Fabr. Latr. Lam. Bosc, Leach.

*External double palpi* with the internal edge of the first joint not dilated. Shell ovate.

*Rostrum* acuminate, acute, with four spines on each side. *Anterior legs* compressed. Abdomen with the sides of the segments obtuse. Tail with the intermediate lamella triangular, the tip emarginate, the apex of the laciniæ rounded. *Interior antennæ* with the first joint of the peduncle trispinous.

Second joint of the internal footstalk of the external double palpi longer than the first.

Sp. 1. *Fabricii.* Anterior legs granulate-spinose; hands externally subserrated; wrists and arms internally spinose. Plate XXI.

Galathea Fabricii. *Leach, Trans. Linn. Soc.* xi. *Crustacea.* 340.

Second joint of the internal footstalk of the external double palpi shorter than the first.

Sp. 2. *Spinigera.* Anterior legs subgranulate squamose, above and on each side spinose; arms externally without spines.

Astacus strigosus. *Penn. Brit. Zool.* iv. 18. pl. 14.

Cancer (astacus) strigosus. *Herbst,* tab. 26. f. 2.

Galathea strigosa.

*Fabr. Ent. Syst.* ii. 471.—Suppl. 414.

*Latr. Gen. Crust. et Insect.* i. 49.

*Leach, Edinb. Encycl.* vii. 398.

Galathea spinigera.

*Leach, Malac. Podoph. Brit.* tab. 28. B.

Rostrum elongate, spiniform, the base on each side bispinose. Anterior pair of legs subcylindric. Abdomen with the sides of the segments acute. Tail with the intermediate lamella transverse-quadrate, the apex subemarginate. Interior antennæ with the first joint of the peduncle four-spined. (External double palpi with the first joint of the internal footstalk longer than the second.)

Sp. 2. *Rugosa.* Anterior legs spinose, especially internally; abdomen with the second segment anteriorly with six, the third with four spines.

Astacus Bamflus. *Pennant, Brit. Zool.* iv. 17. pl. 27.

Galathea rugosa.

*Fabr. Suppl. Ent. Syst.* 415.

*Bosc, Hist. Nat. des Crust.* ii. 87.

*Latr. Hist. Nat. des Crust. et des Insect.* vi. 199. 2.

Cancer rugosus. *Gmelin, Syst. Nat.* i. 2985.

Galathea longipeda. *Lam. Syst. des Animaux Vert.* 158.

Galathea Bamflus. *Leach, Edinb. Encycl.* vii. 398.

Galathearugosa. *Leach, Malac. Podoph. Brit.* tab. 29.

*Trans. Linn. Soc.* xi. 341.

Inhabits the European Ocean and Mediterranean Sea. It is very rare in Britain, but has been found on the Bamffshire coast and in the Plymouth Sound.

**Division V.**

Gen. 51. *Gebia.* Leach.

Two anterior legs equal, subdidactyle, with the thumb short. *Interior antennæ* with an elongate peduncle, the second joint shortest, the third largest and cylindric. *External double palpi* with the third joint of the internal footstalk shortest. Tail with broad lamellæ, the exterior ones costated, the middle one quadrate.

Sp. 1. *Deltauira.* Abdomen with the back membranaceous; tail with the apex of the exterior lamella dilated, and somewhat rounded; interior one truncate and formed like the Greek Delta.

Gebia deltauira. *Leach, Trans. Linn. Soc.* xi. 342.

Inhabits beneath the sand on the southern coast of Devonshire, and is found by digging to the depth of two or three feet.

Gen. 52. *Callianassa.* Leach.

Four anterior legs didactyle; anterior pair largest, very unequal; second pair less; third pair monodactyle; fourth and fifth pairs spurious. *Internal antennæ* with an elongate biarticulate peduncle, the Crustacea, second joint longest. External double palpi with the second joint of the internal footstalk largest and compressed. Tail with broad lamellae; the middle process elongate-triangular, with the apex rounded.

The thorax anteriorly abruptly subacuminate; the rostriform process divided from the shell by a suture. Anterior pair of legs very much compressed, the hand articulated. The larger leg, with the base of its wrist furnished with a curved process.

Sp. 1. Subterranea. Shell with the rostriform process with one longitudinal ridge, the point rounded. Cancer astacus subterraneus.

Montagu, Trans. Linn. Soc. xi.

Callianassa subterranea.

Leach, Edin. Encycl. vii. 400.

Trans. Linn. Soc. xi. 343.

This animal lives beneath the sand on the sea-shore. It was first described by Montagu, who found it by digging in a sand-bank in the estuary of Kingsbridge, on the southern coast of Devon.

Gen. 53. Axis. Leach.

Four anterior legs didactyle; anterior pair largest, and somewhat unequal; third, fourth, and fifth pairs, furnished with a compressed claw. Interior antennae with a three-jointed peduncle, the first joint longest. External double palpi with the two first joints somewhat large and equal. Tail broad, the intermediate lamella elongate-triangular.

Sp. 1. Stirynchus. Rostrum margined, the middle carinated; thorax behind the rostrum with two elevated abbreviated lines notched behind.

Axis Stirynchus. Leach, Trans. Linn. Soc. xi. 343.

Inhabits the British Sea.

Division VI.

Gen. 54. Astacus.

Fabr. Lam. Latr. Bosc, Penn. Leach.

Eyes subglobose, not thicker than their peduncles. Exterior antennae with the first joint of the peduncle furnished with a spiniform squama that does not reach to the apex of the peduncle.

The coxae of the third pair of legs of the female, of the fifth pair of the male, perforated. These perforations are for the passage of the semen and of the eggs, and although placed differently in other genera, yet they serve the same functions.

Abdomen with the sides of its segments obtuse.

Astaci Marini.

Sp. 1. Gammarus. Rostrum on each side with four teeth, and with one on each side of its base. Cancer gammarus. Linn. Syst. Nat. i. 1050.

Astacus gammarus. Penn. Brit. Zool. iv. 9. pl. 10.

Astacus marinus.

Fabr. Suppl. Ent. Syst. 406.

Latr. Gen. Crust. et Insect. i. 51.

Astacus gammarus.

Leach, Edin. Encycl. vii. 398.

Trans. Linn. Soc. xi. 344.

This species, which is the common lobster of our markets, inhabits deep clear water at the foot of rocks which hang over the sea. They breed during the early summer months, and are very prolific, Baxter having counted no less than 12,444 eggs under the abdomen. In warm weather they are very active; they have the power of springing backward in Crustacea, the water to a most astonishing distance into their holes in the rocks, as has been repeatedly observed by naturalists of credit. Their food consists of dead animal matter, and, it is said, also of sea-weeds. The female is stated to deposit her eggs in the sand, but the young state is not known.

The common lobster inhabits the European Ocean. It is found in very great abundance in the north of Scotland, but it is much more common on the coast of Norway, from whence the London markets are for the most part supplied.

Aristotle has very distinctly described this species under the name aeraces.

Abdomen with the sides of its segments sharp.

Astici Fluvialites.

Sp. 1. Fluvialitis. Rostrum laterally dentated, the base with one tooth on each side.

Cancer astacus. Linn. Syst. Nat. i. 1051.

Astacus astacus.

Penn. Brit. Zool. iv. 18. pl. 15. fig. 27.

Astacus fluvialitis.

Fabr. Suppl. Ent. Syst. 406.

Latr. Gen. Crust. et Insect. i. 51.

Leach, Edin. Encycl. vii. 400.

Trans. Linn. Soc. xi. 344.

Gen. 55. Nephrops. Leach.

Eyes reniform, abruptly much thicker than their peduncles. Exterior antennae with the first joint of their peduncle furnished at its apex with a squama, which is produced beyond the apex of the peduncle.

The coxae of the third pair of legs of the female, of the fifth pair of the male, perforated.

Sp. 1. Norwegicus. Abdomen with hairy areoles; shell somewhat spiny in front.

Cancer Norwegicus. Linn. Syst. Nat. i. 1053.

Astacus Norwegicus.

Penn. Brit. Zool. iv. 17. pl. 12. fig. 24.

Inhabits the northern parts of Europe. It is found in the Frith of Forth during the summer months, often attaching itself to the lines of the fishermen. Colour, when living, flesh red. Fabricius, Bosc, and Latreille, cannot have seen this animal, as they all describe it as having four, instead of six didactyle legs.

Division VII. Subdivision I.

Gen. 56. Crangon. Fabr. Latr. Bosc, Leach.

Anterior pair of legs largest, with a compressed monodactyle hand, the rest simple; the second and third pair more slender, the fourth and fifth thicker.

Sp. 1. Vulgaris. Thorax behind, and on each side of the rostrum, unispinose.

Cancer crangon. Linn. Syst. Nat. i. 1052.

Astacus crangon. Penn. Brit. Zool. iv. 20.

Crangon vulgaris. Fabr. Suppl. Ent. Syst. 410.

Latr. Gen. Crust. et Insect. i. 55.

Bosc, Hist. Nat. des Crust. ii. 96.

Leach, Edin. Encycl. vii. 401. pl. 21. fig. 5.

Trans. Linn. Soc. xi. 346.

Inhabits the sandy coasts of the European Ocean, often entering estuaries, especially during the breeding season. It is the common shrimp of our markets. Division VII. Subdivision 2.

Gen. 57. Atya. Leach.

Four anterior legs equal, the last joint cloven; third pair largest, unequal, with a simple claw; fourth and fifth pairs simple, terminated by a simple claw. Tail broad, the intermediate lamella with its extremity subacuminate, rounded.

Sp. 1. Scabra. Rostrum carinated trifid, the middle tooth longest; six hinder legs rough. Plate XXI. Atya scabra. Leach, Trans. Linn. Soc. xi. 345.

Gen. 58. Processa. Leach.

Anterior pair of legs, with one side didactyle, the other armed with a simple claw; second pair unequal, didactyle, slender; one very long, with the wrist and fore-arm many-jointed; the other shorter, with the wrist many-jointed; other legs terminated by simple claws.

Sp. 1. Canaliculata. Base of the rostrum with one tooth; intermediate lamella of the tail longitudinally canaliculated.

Processa canaliculata. Leach, Malac. Podoph. Brit. tab. 41.

The thighs of the third and fourth pairs of legs are spinulose beneath; at the base of the rostrum there is an elevation dividing it from the thorax.

The above species, which forms the type of the genus, was discovered at Torcross, on the southern coast of Devon, by Montagu.

Division VII. Subdivision 3.

Gen. 59. Pandalus. Leach.

Anterior pair of legs adactyle; second pair didactyle, unequal. External double palpi with the last joint of the internal footstalk longer than the preceding joint.

Sp. 1. Annulicornis. Rostrum ascending, many-toothed, apex notched; inferior antennae annulated with red, and internally spinulose.

Pandalus annulicornis. Leach, Malac. Podoph. Britann. tab. 40.

Trans. Linn. Soc. xi. 346.

Gen. 60. Hippolyte. Leach.

Four anterior legs didactyle. External double palpi with the last joint of the internal footstalk shorter than the preceding joint.

Sp. 1. Varians. Rostrum straight, with two teeth above and below; shell above and beneath the eyes with one spine.

Hippolyte varians. Leach, Trans. Linn. Soc. xi. 347.

Inhabits the rocky shores of southern Devonshire. It varies much in colour, being often found red, green, and blueish-green.

Gen. 61. Alpheus. Fabr. Latr. Bosc, Leach.

Four anterior legs didactyle. External double palpi with the last joint of the internal footstalk three times longer than the preceding joint.

Sp. 1. Spinus.

Cancer spinus. Sowerby, Brit. Miscel.

Leach, Trans. Linn. Soc. xi. 347.

Inhabits the Scottish Ocean.

Division VII. Subdivision 3.

Gen. 62. Perseus. Fabr. Latr. Bosc, Leach.

Six anterior legs didactyle. External double palpi with five exerted joints, the last of which is obtuse.

Sp. 1. Trisulcatus. Thorax trisulcated behind; rostrum descending, multidentate above.

Perseus trisulcatus.

Leach, Trans. Linn. Soc. xi. 347.

Inhabits the Welsh Sea.

Division VII. Subdivision 4.

Gen. 63. Palemon. Fabr. Latr. Bosc, Leach.

Four anterior legs didactyle; anterior pair smaller than the second pair. External double palpi with the last joint shorter than the preceding joint.

Sp. 1. Serratus (common Prawn). Rostrum ascending above, with from six to eight teeth; the apex emarginate; below with from four to six teeth.

Astacus serratus.

Penn. Brit. Zool. iv. 19. pl. 16. f. 28.

Cancer (astacus) squilla. Herbst, ii. 55. tab. 27. f. 1.

Palemon squilla.

Latr. Gen. Crust. et Insect. i. 54.

Leach, Edinb. Encycl. vii. 401.

Palemon serratus.

Leach, Trans. Linn. Soc. xi. 348.

Variety a Rostrum with six teeth above.

Subvariety 1. Rostrum beneath with four teeth.

Subvariety 2. Rostrum beneath with five teeth.

Variety b Rostrum above with seven teeth.

Subvariety 1. Rostrum beneath with four teeth.

Subvariety 2. Rostrum beneath with five teeth.

Subvariety 3. Rostrum beneath with six teeth.

Variety c Rostrum with eight teeth above.

Subvariety 1. Rostrum beneath with four teeth.

Subvariety 2. Rostrum beneath with five teeth.

Subvariety 3. Rostrum beneath with six teeth.

Although all the above varieties are common, yet b occurs most frequently. We have seen the upper edge of the rostrum with ten, the lower with five teeth; and both edges with but three teeth. The apex is generally notched above, but in two instances we observed the point to be entire. The situation of the teeth on the upper edge is variable, but in most instances the second tooth is at a greater distance from the first than the rest, which are generally equidistant, and rarely extend far beyond the middle, the rostrum from that part being edentate, with the exception of the emarginate apex.

Herbst, Latreille, and Dr Leach, formerly considered this species as Cancer squilla of Linne, but Dr L. has, since the publication of the error, met with the true C. squilla of that author, and has described it in the eleventh volume of the Transactions of the Linnean Society, p. 348.

Palemon serratus of Fabricius is distinct, and, if his description be correct, it is not even referable to this Genus, he having expressly given, as its specific character ("Antennae posticis bifidis"), hinder antennae bifid; whereas, in his generic character, he has stated these organs to be trifid ("Antennae superiores trifidae").

Gen. 64. Athanas. Leach.

Four anterior legs didactyle. Anterior pair larger than the second pair. External double palpi with the last joint longer than the preceding joint.

Sp. 1. Nitecana. Rostrum straight, and simple. Crustacea. Cancer (astacus) nitescens. Montagu's MSS.

Athanais nitescens.

Leach, Trans. Linn. Soc. xi. 349. Inhabits the southern coast of Devonshire.

Division VIII.

Gen. 65. Mysis. * Latreille, Leach.

Praunus. Leach.

Legs bind; the last joint of the four anterior pairs with the interior lacinia uniarticulate, ovate, compressed; of the other pairs of legs multarticulate. External double palpi with the middle joint of the internal footstalk longest, the first very short.

At the base of the abdomen of the female is situated the external uterus, composed of two valve-like membranes, in which the young ones, just excluded from the egg, live and grow until they become strong enough to take care of themselves.

The animals of this genus swim with their head uppermost, and with their eyes spreading, which gives them a singular and grotesque appearance.

* Intermediate lamella of the tail emarginate.

Sp. 1. Spinulosa. Tail with the intermediate lamella externally spinulose, the apex acutely emarginate; exterior lamellae acuminate, and very broadly ciliated.

Praunus flexuosus. Leach, Edinb. Encycl. vii. 401. Mysis spinulosa. Leach, Trans. Linn. Soc. xi. 350. Inhabits the Frith of Forth near Leith.

Colour when alive, pellucid cinereous. Eyes black, red at their base. Laminae of the external antennae with a black longitudinal line and spots. A clouded spot on each side of the hinder part of the thorax, and another above the legs. Every segment of the body most beautifully marked with a reddish-rust-coloured spot, disposed in an arborescent form; tail fin spotted with the same colour, mixed with black. Pouch of the female with two rows of fuscos-black spots. Under side of the abdomen regularly mottled with rufous black.

It was observed with young from the middle of June to the middle of July. The females are one-third more abundant than the males. Length an inch and a quarter.

Sp. 2. Fabricii. Intermediate lamella of the tail obtusely notched; exterior lamellae with rounded points.

Mysis Fabricii. Leach, Trans. Linn. Soc. xi. 350. Inhabits the Greenland Sea, affording the principal sustenance of the great northern whale (Balaena mysticetus.)

** Intermediate lamella of the tail entire.

Sp. 3. Integra. Praunus integer.

Leach, Edinb. Encycl. vii. 401. Mysis integra. Trans. Linn. Soc. xi. 350. Inhabits brackish pools of water left by the tide at Loch Ranza in the isle of Arran. Common in the month of August with young.

Length one-third of an inch. Females more abundant than the males. Colour whilst living pellucid, cinereous, spotted with black and reddish brown.

Division IX.

Gen. 66. Nebalia. Leach.

Thorax anteriorly with a moveable rostrum. Anterior pair of legs longest, simple; other pairs equal, approximate with the last joint bifid. Antennae two, inserted above the eyes, the last joint bifid and multarticulate.

Sp. 1. Herbstii. Gray or cinereous-yellowish; eyes black.

Cancer bipes.

Oth. Fabr. Fa. Grön. No. 223. fig. 2. Herbst, ii. tab. 24. fig. 7. Mysis bipes.

Latr. Hist. Nat. des Crust. et des Insect. vi. 285. Monoculus rostratus.

Montagu, Trans. Linn. Soc. xi. 14. tab. 2. f. 5. Nebalia Herbstii.

Leach, Zool. Miscel. i. 100. t. 44. —-, Trans. Linn. Societ. xi. 351. Inhabits the European Ocean; it is common beneath stones lying on black mud, on the southern coast of Devon.

Macrouros Genera of uncertain situation.

Gen. 67. Albunea.

Dold. Fabr. Bosc, Lam. Latr. Leach.

Internal antennae with their peduncles shorter than the two sets by which they are terminated. Legs ten; anterior pair with monodactyle hands, the thumb uncinate; hinder legs minute, spurious, filiform; the other legs terminated by a compressed sulcate joint. Tail not fan-shaped.

Sp. 1. Symnista. Shell anteriorly serrated.

Cancer symnista. Linn. Syst. Nat. i. 1053. Albunea symnista.

Fabr. Ent. Syst. Suppl. 397. Latr. Gen. Crust. et Insect. i. 44. Leach, Edinb. Encycl. vii. 396. Inhabits the Indian Ocean.

Gen. 68. Remipes. Latreille, Leach.

Internal antennae with their peduncles shorter than the two sets by which they are terminated. Legs ten; the three hinder pairs, alike, and formed for swimming; second pair longer than the first, terminated by a conic compressed joint. Tail not fan-shaped.

Sp. 1. Testudinarius.

Remipes testudinarius.

Latr. Gen. Crust. et Insect. i. 45. Leach, Edinb. Encycl. vii. 396. Inhabits the New Holland Seas.

Gen. 69. Hippa. Fabr. Lam. Latr. Bosc, Leach.

Emerita. Gromortius.

Internal antennae, with their peduncles shorter than the two sets by which they are terminated. Legs ten; anterior pair adactyle, second and third pair with the last joint lunate; fourth pair with the last

---

* In the Edinburgh Encyclopaedia Dr Leach gave the Genus Mysis solely on the authority of the generally accurate Latreille, who formed the genus without any actual examination of its characters; and as he described but twelve legs, and misplaced it in the system, Dr Leach holds himself justified in having described the same Genus under the new name (Praunus), which we have now rejected for that given by Latreille. Crustacea, joint trigonal; hinder pair minute, filiform, spurious.

Tail not fan-shaped.

Sp. 1. Emeritus. Tail inflexed, the last joint ovate.

Hippa emeritus,

Fabr. Ent. Syst. Suppl. 370.

Latr. Gen. Crust. et Insect. i. 45.

Leach. Edinb. Encycl. vii. 396.

Inhabits the Indian Ocean.

Gen. 70. Thalassina. Latreille, Leach.

Internal antennae terminated by two setæ, and inserted in the same horizontal line with the external ones. Legs ten, the two anterior pairs didactyle. Tail composed of five plates resembling a fan.

Sp. 1. Scorpionoides.

Thalassina scorpionoides.

Latr. Gen. Crust. et Insect. i. 52.

Leach, Edinb. Encycl. vii. 400.

Gen. 71. Squilla.

Fabr. Bosc, Lam. Latr. Leach, &c.

Internal antennæ with three setæ. Legs fourteen; anterior pair largest, monodactyle, the thumb much spined within; second, third, and fourth pairs with a monodactyle hand, the thumb being crooked and simple; the fifth, sixth, and seventh pairs spurious.

Sp. 1. Mantis. Body above with many elevated longitudinal lines; thumbs with six teeth.

Squilla mantis.

Latr. Gen. Crust. et Insect. i. 55.

Leach, Edinb. Encycl. vii. 402.

Inhabits the Mediterranean Sea.

Gen. 72. Zöe. Latr. Leach.

Zoea. Bosc.

Eyes two, sessile, inserted one on each side of the head; Rostrum perpendicular, of the length of the thorax. Thorax somewhat ovate; shell diaphonous, with the back produced into a spine. Legs obscure and short, with the exception of the hinder ones, which are elongate, and formed for swimming. Tail as long as the thorax, and generally bent under it, composed of five joints, the first four very narrow, the last larger, lunate and spinulous.

Sp. 1. Pelagica. Spine of the back twice the length of the thorax, and bent backwards.

Zoea pelagica.

Bosc, Hist. Nat. des Crust. ii. 135. pl. 15. f. 3, 4.

Latr. Gen. Crust. et Ins. i. 21.

Leach, Edinb. Encycl. vii. 389.

Inhabits the Atlantic Ocean. Was first described by Bosc in the above quoted work.

Legion II. EDRIOPHTHALMA.

The Malacostraca Edriophthalma, or at least a greater part of them, were placed amongst the Macroura by Latreille, who considered them as forming a particular family of that order. Had he examined the following new and curious genera, he would doubtless have formed a very different opinion. Many of the genera he even included amongst the Arachnides, as shall be shown hereafter.

Synopsis and distribution of the Genera.

Section I.

Body laterally compressed. Legs fourteen. Antennæ two, inserted one on each side of the front of the head. (Tail furnished with styles.)

Genus 1. Phronyma.

Section II.

Body laterally compressed. Legs fourteen, with lamelliform coxa. Antennæ four, inserted by pairs. (Tail furnished with styles.)

Division I. Antennæ four-jointed, the last segment composed of many little joints; the upper ones very short.

Genus 2. Talitrus.

3. Orchestia.

Division II. Antennæ four-jointed, the last joint composed of several little joints; upper ones rather shortest.

Genus 4. Atylus.

Division III. Antennæ three-jointed, the last joint composed of several little joints; upper ones longest.

Genus 5. Dexamene.

6. Leucothoe.

Division IV. Antennæ four-jointed, the last segment composed of several little joints; upper ones longest.

Subdivision 1. Four anterior legs monodactyle, second pair with a much dilated compressed hand.

Genus 7. Melita.

8. Mera.

Subdivision 2. Two anterior pair monodactyle and alike.

Genus 9. Gammarus.

10. Amphithoe.

11. Pherus.

Division V. Antennæ four-jointed, under ones longest, leg-shaped. (Four anterior legs monodactyle.)

Subdivision 1. Second pair of legs with a large hand.

Genus 12. Podocerus.

13. Jassa.

Subdivision 2. Second pair of legs with a moderate-sized hand.

Genus 14. Corophium.

Section III.

Body depressed. Antennæ four. Legs fourteen.

A. Tail without appendices.

Division I. Body with all the segments bearing legs.

Subdivision 1. Body linear.

Genus 15. Proto.

16. Caprella.

Subdivision 2. Body broad.

Genus 17. Larunda.

Division II. Body with all the segments not bearing legs.

Genus 18. Idotea.

19. Stenosoma.

B. Tail on each side, with one or two appendices.

Division III. Antennæ inserted in nearly the same horizontal line.

Genus 20. Anthura.

Division IV. Antennæ inserted in pairs, one above the other.

Subdivision 1. Tail with one lamella on each side.

Genus 21. Campecopæa.

22. Nesa.

Subdivision 2. Tail with two lamellæ on each side.

* Superior antennæ with a very large peduncle. Claws bifid. Genus 23. Cymodice. 24. Dynamene. 25. Spirocoma.

** Superior antennae with a very large peduncle. Claws single.**

Genus 26. Aega. *** Superior antennae with a moderate peduncle.**

Genus 27. Eurydice. 28. Limnoria. 29. Cymothoa. C. Tail terminated with two setæ.

Division V. Genus 30. Apseudes. D. Tail furnished with styles.

Division VI. Interior antennae distinct. Subdivision 1. Styles of the tail exserted. Anterior legs monodactyle.

Genus 31. Janira. 32. Asellus.

Subdivision 2. Styles of the tail not exserted. Anterior legs simple.

Genus 33. Jera.

Division VII. Interior antennae not distinct. Subdivision 1. Styles of the tail double, with a double footstalk.

Genus 34. Licia.

Subdivision 2. Styles of the tail four, the lateral ones biarticulate.

* Body not capable of contracting into a ball. a. External antennae eight-jointed.

Genus 35. Philoscia. 36. Oniscus.

b. External antennae with seven joints.

Genus 37. Porcellio. ** Body contractile into a ball.

Genus 38. Armadillo. Genus of uncertain situation.

Genus 39. Bopyrus.

Section I. Gen. 1. Phronyma. Latreille, Leach, Lamarck. Head large, nutant; antennae biarticulate, the first joint small. Thorax seven-jointed, all its segments bearing legs. Legs compressed, two anterior pairs with the antepenultimate joint furnished at its point with a foliaceous process; the penultimate joint with the point bifid and terminated with a small claw; third and fourth pairs simple, longer, somewhat thicker, terminated by a bent claw; fifth pair large, very long, thicker, didactyle; the first joint gradually thickened towards its point; the second subtrigonate; the third ovate, and abruptly narrowed at its base; the last narrowed at its base; the fingers curved, and internally furnished each with one tooth; sixth and seventh pairs simple, terminated with a nearly strait claw. Abdomen triarticulate, each segment, on each side, with a double appendice, placed on a peduncle. Tail biarticulate, the first joint on each side furnished with a biarticulate process, terminated by two styles; second joint with four processes, each terminated by two styles; the inferior processes biarticulate, the superior triarticulate.

Sp. 1. Sedentaria. Fifth legs with the apex of the thumb and base of the finger internally denticulated. Cancer sedentarius. Forsk. Fn. Arab. 95.

Phronima sedentaria. Latr. Gen. Crust. et Insect. i. 57. Leach, Edinb. Encycl. vii. 403—433. Trans. Linn. Soc. xi. 355.

Cancer (gammarellus) sedentarius. Herbst, ii. 136. tab. 37. fig. 8.

Inhabits the Mediterranean Sea and Zetland Sea, residing in a cell composed of a gelatinous substance, open at each extremity, where it sits in an incurved posture.

The only specimen of this most interesting, rare, and curious animal, that has come under our inspection, was sent to us by the Reverend Dr J. Fleming, one of our most zealous Naturalists, who found it on the 3rd November 1809, at Burry in Zetland, amongst rejectamenta of the sea.

All authors have erred in giving but ten legs to this animal. Of the parts of the mouth, we can, at present, say nothing.

Section II.

Division I. Gen. 2. Talitrus. Latreille, Bosc, Leach. Four anterior legs in both sexes subequal; monodactyle. Upper antennae shorter than the two first joints of the under ones.

Sp. 1. Locusta. Antennae subtestaceous-rufous, of the male longer than the body, of the female shorter; body cinereous, varied with darker cinereous. Oniscus locusta. Pallas?

Talitrus locusta.

Latr. Gen. Crust. et Insect. i. 58. Bosc, Hist. Nat. des Crust. ii. 152. Leach, Edinb. Encycl. vii. 402.

Astacus locusta. Penn. Brit. Zool. iv. 21.

Cancer (gammarus) saltator. Montagu, Trans. Linn. Soc. xi. 94.

Inhabits the sandy shores of the European Ocean. The specific name Locusta is probably derived from the form of its protruded mouth, which has a general resemblance to the same part in the gryllides.

It has never been observed in the water; it burrows in the sand, and leaps about on the shore.

Talitrus litoralis, described in the seventh volume of the Edinburgh Encyclopedia, is merely the female of T. locusta.

The use of this animal (which is generally denominated sand-hopper) in the economy of nature, appears to be that of contributing to the dissolution of putrid animal and vegetable matter; serving in return as food to the shore-birds, who devour it with avidity.

Gen. 3. Orchestia. Leach. Four anterior legs of the male monodactyle, second pair with a compressed hand; of the female with the anterior pair monodactyle, the second didactyle. Upper antennae not longer than the two first joints of the under ones.

Sp. 1. Littorea.

Cancer gammarus littoreus. Montagu, Trans. Soc. xi. 96. Leach, Edinb. Encycl. vii. 402. pl. 21. fig. 6. Trans. Linn. Soc. xi. 356.

Inhabits many of our shores, and is found at the Division II.

Gen. 4. Atylus. Leach.

Upper antennae with the second joint longer than the third; under ones with the second joint somewhat shorter than the third. Eyes sub-prominent, rounded, inserted in a process on each side of the head, between the antennae. Tail on each side with three double styles, and above with one moveable style on each side.

Sp. 1. Carinatus. Head with the rostrum descending; five last segments of the abdomen carinated, and acutely produced behind.

Gammarus carinatus. Fabr. Ent. Syst. ii. 515. 3.

Atylus carinatus.

Leach, Zool. Miscel. ii. 22. tab. 69.

Trans. Linn. Soc. xi. 357.

The locality is unknown.

Division III.

Gen. 5. Dexamine. Leach.

Four anterior legs sub-equal, monodactyle, furnished with a filiform-subovate hand. Antennae with their first joint shortest. Eyes oblong, not prominent, inserted behind the superior antennae. Tail on each side with three double styles, and above on each side with one moveable style.

Sp. 1. Spinosa, segments of the abdomen behind, produced into spines.

Cancer (gammarus) spinosus.

Montagu, Trans. Linn. Soc. xi. 3.

Dexamine spinosa.

Leach, Edin. Encycl. vii. 483.

Zool. Miscel. ii. 24.

Trans. Linn. Soc. xi. 359.

Inhabits the sea of the western coasts of Britain.

Gen. 6. Leucothoe. Leach.

Anterior pair of legs didactyle, the thumb biarticulate; second pair with a dilated and compressed hand, furnished with a crooked thumb.

Sp. 1. Articulosa.

Cancer articulosus.

Montagu, Trans. Linn. Soc. vii. 71. tab. 6. f. 6.

Leucothoe articulosa.

Leach, Edin. Encycl. vii. 403.

Trans. Linn. Soc. xi. 358.

Inhabits the British Sea, but is very rare.

Division IV. Subdivision I.

Gen. 7. Melita. Leach.

Anterior pair of legs monodactyle; second pair with the thumb inflexed on the palm. Tail on each side with an elongate foliaceous lamella.

Sp. 1. Palmata. Body blackish; antennae and legs annulated with pale colour.

Cancer palmatus. Montagu, Trans. Linn. Soc. vii. 69.

Melita palmata.

Leach, Edin. Encycl. vii. 403.

Trans. Linn. Soc. xi. 358.

Plate XXI.

Inhabits the sea-shore on the Devonshire coast, under stones.

Gen. 8. Mæra. Leach.

Four anterior legs didactyle; thumb of the second pair bent on the side of the hand. Tail with no foliaceous appendices.

Sp. 1. Grossimana.

Cancer gammarus grossimanus.

Montagu, Trans. Linn. Soc. ix. 97. t. 4. f. 5.

Mæra grossimana.

Leach, Edin. Encycl. vii. 403.

Trans. Linn. Soc. xi. 359.

Inhabits the southern coast of Devonshire, beneath stones.

Division IV. Subdivision 2.

Gen. 9. Gammarus. Latreille, Leach.

Superior antennae furnished at the base of the fourth joint with a little jointed seta. Tail above with bundles of spines.

Tail with the superior double styles, having the upper style process very short.

Sp. 1. Aquaticus. Process between the antennae rounded, obtuse.

Gammarus pulex.

Leach, Edin. Encycl. vii. 402—432.

Gammarus aquaticus.

Leach, Trans. Linn. Soc. xi. 359.

Inhabits ponds, ditches, and springs in great plenty. We formerly considered it to be the same with the Gammarus pulex of Latreille and Bosc, but, on examining the subject more closely, we find their figures are those referred to by them, representing the hands much dentated within.

Sp. 2. Marinus. Process between the antennæ sub-acuminate.

Gammarus marinus. Leach, Trans. Linn. Soc. xi. 359.

Inhabits the sea on the southern coast of Devonshire in plenty.

Tail with the superior double styles, having the style processes subequal.

Sp. 3. Locusta. Eyes lunate.

Cancer gammarus locusta.

Montagu, Trans. Linn. Soc. ix. 92.

Gammarus locusta.

Leach, Edin. Encycl. vii. 403.

Trans. Linn. Soc. xi. 359.

Inhabits the British Sea.

Sp. 4. Camylops. Eyes flexuous.

Gammarus camylops.

Leach, Edin. Encycl. vii. 403.

Trans. Linn. Soc. xi. 360.

Inhabits the sea about Loch Ranza, in the Isle of Arran, where we took a single pair of this interesting animal.

Gen. 10. Ampithoe. Leach.

Superior antennae with no seta at the base of their fourth joint. Tail simple above. Hands ovate.

Sp. 1. Rubricata.

Cancer gammarus rubricatus.

Montagu, Trans. Linn. Soc. ix. 99.

Gammarus rubricatus.

Leach, Edin. Encycl. vii. 402.

Ampithoe rubricata.

Leach, Edin. Encycl. vii. 432.

Trans. Linn. Soc. xi. 360.

Inhabits the sea of the southern coast of Devon. Gen. 11. **Pherusa. Leach.** Superior antennae with no seta at the base of their fourth joint. Tail simple above. Hands filiform. Sp. 1. *Fucicola.* Testaceous-cinerous, or gray-cinerous, mottled with reddish. *Pherusa fucicola.* Leach, Edin. Encycl. vii. 432. Trans. Linn. Soc. xi. 360.

Plate XXI. Inhabits fuci on the southern coast of Devonshire.

Division V. Subdivision 1. Gen. 12. **Podocerus. Leach.** Eyes prominent. Four anterior legs monodactyle. Sp. 1. *Variiegatus.* Body varied with red and white. *Podocerus variiegatus.* Leach, Edin. Encycl. vii. 433. Trans. Linn. Soc. xi. 361. Inhabits the southern coast of Devonshire, amongst conveve and corallines.

Gen. 13. **Jassa. Leach.** Eyes not prominent. Four anterior legs monodactyle with oval hands; second pair with its internal edge dentated. Sp. 1. *Pulchella.* Thumb of second pair of legs with its internal edge notched at the base; colour white painted with red. Var. a. Hands of the second pair with an elongate obtuse tooth. Var. b. Hands of the second pair with the internal edge tridentate. *Jassa pulchella.* Leach, Edin. Encycl. vii. 433. Trans. Linn. Soc. xi. 361. Inhabits the sea of southern Devon, amongst fuci.

Division V. Subdivision 2. Gen. 14. **Corophium. Latreille, Leach.** Sp. 1. *Longicornis.* Cancer grossipes. Linn. Syst. Nat. i. 1055. Astacus grossipes. Penn. Brit. Zool. iv. pl. 16. fig. 31. Corophium-longicornis. Latr. Gen. Crust. et Insect. i. 59. Leach, Edin. Encycl. vii. 408—432. Trans. Linn. Soc. xi. 662. Inhabits the coast of the European Ocean. At low tide, it may be observed crawling amongst the mud. It is very common at the mouth of the river Medway, from whence we have received a vast number of specimens.

Section III. A. Division I. Subdivision 1. Gen. 15. **Proto. Leach.** Second, third, and fourth pair of legs appendicated at their bases. To this genus belongs *Squilla pedata,* and probably also *Ventricosa* of Müller, with *Cancer gamma-rus pedatus* of Montagu, which is probably the same with *S. pedata* of Müller. See Transactions of the Linnean Society, Vol. XI. page 6. tab. 11. fig. 6.

Gen. 16. **Caprella. Lamarck, Latr. Bosc, Leach.** Second, third, and fourth pairs of legs not appendicated at their bases; the third and fourth pairs Crustacea, spurious, subgelatinous, and globose.

The animals composing this genus, inhabit the sea, living amongst sertulariae and marine plants, moving geometrically like the larvae of the Phalaridea.

The specific characters may be taken from the number and situation of the spines on the head and back, form of the second pair of legs, &c. Sp. 1. *Phasma.* Hands of the second pair of legs narrow, their internal edge acutely notched backwards; back anteriorly with three spines, turning forwards. *Cancer phasma.* Montagu, Trans. Linn. Soc. vii. 66. tab. 6. f. 3. Inhabits the southern coast of Devon. *Astacus atomos* of Pennant, and *Squilla lobata* of Müller belong to the genus Caprella, of which we have some unpublished species.

A. Division I. Subdivision 2. Gen. 17. **Larunda. Leach.** Cyamus. Latreille, Bosc. Panope. Leach. Antennae four-jointed, upper ones longest. Legs compressed, with strong claws; the third and fourth pairs elongate, spurious, cylindric, without claws; the two anterior pairs monodactyle. External uterus, or pouch of the female, composed of four valves. Sp. 1. *Ceti.* Bases of the third and fourth pairs of legs with processes resembling the figure 6; the hands of the second pair of legs anteriorly with three obtuse teeth. Plate XXI. *Oniscus ceti.* Linn. Syst. Nat. i. 1060. Pull. Spec. Zool. ix. 4. f. 14. *Squille de la baleine.* De Geer, Mém. sur les Insect. vii. pl. 42. f. 6. 7. *Pyurogonum ceti.* Fabr. Suppl. Ent. Syst. 570. Cyamus ceti. Latr. Gen. Crust. et Insect. i. 60. Panope ceti. Leach, Edinb. Encycl. vii. 404. *Larunda ceti.* Leach, Trans. Linn. Soc. xi. 364. Inhabits whales, and, according to Latreille, it is also found on some species of the genus Scomber. By the Greenland fishermen it is termed the whale-louse.

Division II. Gen. 18. **Idotea. Fabr. Latr. Bosc, Leach.** Asellus. Olivier, Lamarck. Entomon. Klein. External antennae half the length of the body, or less; the third and fourth joints equal. Body ovate. Sp. 1. *Pelagica.* Body linear-oval; tail rounded, the middle with a very obsolete tooth; antennae one third of the length of the body. *Idotea pelagica.* Leach, Trans. Linn. Soc. xi. 365. Inhabits the Scottish Seas.

Mr Stevenson sent us this species from the Bell-Rock, and afterwards procured for us a large log, perforated by Limnoria terebrans, which contained a vast number of them in the deserted cavities formed by that animal. It was taken in the Firth of Forth by the Rev. Dr Fleming, in whose collection there are specimens. Colour when alive ash-gray or fuscous, speckled with darker colour, and often variegated or mottled with white spots; legs pale.

The female seems to be very rare, as amongst four hundred specimens of the animal, one only of that sex was found.

Length one inch and a quarter.

Gen. 19. **Stenosoma.** Leach.

*External antennae* as long as the body, the third joint longer than the fourth. *Body* linear.

Sp. 1. **Lineare.** Last segment of the tail somewhat narrowed at its base, and dilated towards its apex, which is truncate and notched.

Oniscus linearis.

*Penn. Brit. Zool.* iv. pl. 18. fig. 2.

Idotea hectarica. Leach, Edin. Encycl. vii. 404.

Stenosoma hectaricum.

Leach, Edin. Encycl. vii. 433.

Stenosoma lineare.

Leach, Trans. Linn. Soc. xi. 366.

Inhabits the European Ocean. It sometimes occurs in the Firth of Forth, and amongst the Hebrides.

B. Division III.

Gen. 20. **Anthura.** Leach.

*Antennae* short subequal, inserted one after another in the same horizontal line, the internal ones a little longest. *Body* linear. *Tail* with the last joint but one very short, the last elongate, narrower, with two elongate lamellae on each side.

Sp. 1. **Gracilis.** Lateral processes of the tail obliquely truncated.

Oniscus gracilis.

Montagu, Trans. Linn. Soc. ix. tab. 5. fig. 6.

Anthura gracilis.

Leach, Edin. Encycl. vii. 404.

Trans. Linn. Soc. ix. 366.

B. Division IV. Subdivision 1.

Gen. 21. **Campecofea.** Leach.

*Tail* with its last segment furnished on each side with a compressed, curved, appendage. *Body* six-jointed, the last joint of the same size with the others. *Antennae* setaceous, upper ones longest, their peduncle biarticulate; the space between the antennae very great. *Anterior clavus* bifid (the others I have not seen).

Sp. 1. **Hirsuta.** Brown, the last joint of the body with a few faint blueish spots.

Oniscus hirsutus.

Montagu, Trans. Linn. Soc. vii. t. 6. f. 8.

Campecopea hirsuta.

Leach, Edin. Encycl. vii. 405.

Trans. Linn. Soc. xi. 367.

Inhabits the southern coast of Devonshire; but is rather rare. Length one eighth of an inch.

Gen. 22. **Nesia.** Leach.

*Tail* on each side of the last segment with a strait, subcompressed process attached to a peduncle. *Body* six-jointed, the last joint largest. *Antennae* setaceous, subequal; upper ones with a very large biarticulated peduncle, the first joint largest; space between the antennae easily to be discerned. *Clavus* bifid.

Sp. 1. **Bidentata.** Last segment of the body armed with two spines or teeth; colour cinereous, faintly streaked with blue, or reddish.

Oniscus bidentatus.

Adams, Trans. Linn. Soc. v. s. t. 2. f. 3.

Nesia bidentata.

Leach, Edin. Encycl. vii. 405.

Trans. Linn. Soc. xi. 367.

Inhabits the coasts of Wales and Devonshire.

Division IV. Subdivision 2.

Gen. 23. **Cymodice.** Leach.

*Eyes* touching the anterior margin of the first segment of the body. *Body* seven-jointed. *Tail* at the base on each side with two sub-compressed but not foliaceous appendages, the exterior ones largest; the apex of the tail notched with a lamella in the centre. *Clavus* bifid.

Sp. 1. **Truncata.** Apex of the tail truncate.

Oniscus truncatus.

Montagu's MSS.

Leach, Trans. Linn. Soc. xi. 303.

Edin. Encycl. vii. 433.

This species is very rare, and has been found but three times on the southern coast of Devonshire.

Gen. 24. **Dynamene.** Leach.

*Eyes* not reaching to the anterior margin of the first segment of the body. *Body* seven-jointed. *Tail* with two equal foliaceous appendages on each side of its base; the apex notched. *Clavus* bifid.

Dynamene.

Leach, Edinb. Encycl. vii. 405.

Trans. Linn. Soc. xi. 303.

Gen. 25. **Sphaeroma.** Latreille, Leach.

*Eyes* not reaching to the anterior margin of the first segment of the body. *Body* seven-jointed. *Tail* with its apex entire; the base on each side with two equal foliaceous appendages. *Clavus* bifid.

Sp. 1. **Serrata.** Body smooth, unarmed; tail very smooth on each side, obliquely truncated; lamellae elliptic, acute, the external ones externally serrated.

Oniscus globator.

Pall. Spec. Zool. Fasc. ix. tab. 4. fig. 18.

Cymothoea serrata.

Fabr. Ent. Syst. ii. 510.

Sphaeroma cinerea.

Latr. Gen. Crust. et Insect. i. 65.

Sphaeroma serrata.

Leach, Edinb. Encycl. vii. 405.

Trans. Linn. Soc. xi. 303.

B. Division IV. Subdivision 2.

Gen. 26. **Æga.** Leach.

*Eyes* large, granulated, oblong, oblique, marginal. *Tail* with its appendages foliaceous.

Sp. 1. **Emarginata.** Tail with the last joint acuminate; the interior lamella internally obliquely truncated, externally emarginated. Plate XXI.

Æga emarginata.

Leach, Trans. Linn. Soc. xi. 370.

B. Division IV. Subdivision 2.

Gen. 27. **Eurydice.** Leach.

*Eyes* distinct, simple, lateral. *Head* as broad as the first segment of the body.

Sp. 1. **Pulchra.** Tail with the last joint semioval; body cinereous, variegated with black.

Gen. 28. **Limnoria.** Leach.

*Head* as broad as first segment of the body. *Eyes* granulated. Crustacea. Sp. 1. Terebrans. Body cinereous; eyes pitchy-black.

Limnoria terebrans.

Leach, Edinb. Encycl. vii. 433. Trans. Linn. Soc. xi. 370.

Inhabits the British Ocean, perforating buildings of wood, piles, &c. It is common at the Bell-Rock, and on the coasts of Suffolk and Yorkshire. It generally produces seven young ones.

Gen. 29. Cymothoa. Fabr. Dald. Leach, &c. Head narrow and small. Eyes obsolete; Body with the first segment notched to receive the head.

Sp. 1. Oestrum. Cymothoa oestrum. Fabr. Ent. Syst. ii. 505.

C. Division V.

Gen. 30. Apsuedes. Leach. Sp. 1. Talpa. Shell anteriorly sharp, rostriform, with three excavated longitudinal lines.

Cancer gammarus talpa.

Mont. Tr. Linn. Soc. ix. t. 4. f. 6.

Apsuedes talpa.

Leach, Edinb. Encycl. vii. 404. Trans. Linn. Soc. xi. 372.

Inhabits the British Sea.

D. Division VI. Subdivision 1.

Gen. 31. Janira. Leach. Claws bifid. Eyes moderate lateral-subvertical. Internal antennae shorter than the peduncle of the external ones.

Sp. 1. Maculosa. Body cinereous, maculated with fuscosus.

Oniscus maculosus. Montagu's MSS.

Janira maculosa.

Leach, Edinb. Encycl. vii. 434. Trans. Linn. Soc. xi. 373.

Inhabits the southern coast of Devonshire, amongst marine plants.

Gen. 32. Asellus. Geoffroy, Olivier, Latreille, Bosc, Leach.

Entomon. Klein. Claws simple. Eyes minute, lateral. Interior antennae of the length of the setiferous joint of the exterior ones.

Sp. 1. Aquaticus. Colour cinereous, either spotted with grey or whitish.

Oniscus aquaticus.

Linn. Syst. Nat. i. 1061.

Aselle d'eau douce.

Geoff. Hist. des Insect. ii. 672. pl. 22. fig. 2.

Squille asselle.

De Geer, Memo. sur les Insect. vii. 496. pl. 31. fig. 1.

Aselle ordinaire.

Latr. Hist. Nat. des Crust. et des Insect. vi. 359.

Asellus vulgaris.

Bosc, Hist. Nat. des Crust. ii. 170. pl. 15. fig. 7.

Latr. Gen. Crust. et Insect. i. 63. Leach, Edin. Encycl. vii. 404.

Idotea aquatica. Fabr. Suppl. Ent. Syst. 303.

Entomon hieroglyphicum. Klein, Dub. fig. 5.

Asellus aquaticus.

Leach, Trans. Linn. Soc. xi. 373.

Inhabits ponds and ditches, and is generally considered a sign of the purity of the water.

Division VI. Subdivision 2.

Gen. 33. Jæra. Leach. Eyes moderately large, situated between the sides and the vertex of the head.

Sp. 1. Albifrons. Cinereous; front whitish.

Oniscus albifrons. Montagu's MSS.

Jæra alifrons.

Leach, Edinb. Encycl. vii. 434. Trans. Linn. Soc. xi. 379.

Inhabits marine plants, and beneath stones, on the southern coast of Devon.

Division VII. Subdivision 1.

Gen. 34. Ligia. Fabricius, Latreille, Bosc, Leach. External antennæ with the last joint composed of several other joints.

Sp. 1. Oceanica. Antennæ as long as the body; back subscabrose.

Ligia oceanica.

Fabr. Suppl. Ent. Syst. 301. Leach, Edinb. Encycl. vii. 406.

Ligia scopulorum.

Leach, Edinb. Encycl. vii. 406.

Oniscus oceanicus.

Linn. Syst. Nat. i. 1061.

Inhabits the rocky shores of the European Ocean. The last joint of the antennæ varies much in the number of its segments, even in the same individual.

Division VII. Subdivision 2. a.

Gen. 35. Philoscia. Latreille, Leach. External antennæ with their bases naked. Tail abruptly narrower than the body.

Sp. 1. Muscorum. Body variegated; sometimes plain brick-red.

Oniscus muscorum. Scop. Ent. Carn. 1145.

Oniscus sylvestris. Fabr. Ent. Syst. iv. 397.

Philoscia muscorum.

Latr. Gen. Crust. et Insect. i. 69. Leach, Edinb. Encycl. vii. 406.

Inhabits France, Germany, and England, under stones and mosses.

Gen. 36. Oniscus of authors. Antennæ inserted beneath the anterior margin of the head, on a prominent part.

Sp. 1. Asellus. Above obscure cinereous, rough; the sides and a series of dorsal spots yellowish.

Oniscus asellus.

Linn. Syst. Nat. i. 1061. Latr. Gen. Crust. et Insect. i. 70. Leach, Edinb. Encycl. vii. 406. Trans. Linn. Soc. xi. 375.

Oniscus murarius.

Fabr. Suppl. Ent. Syst. 300.

Inhabits rotten wood and old walls throughout the greater part of Europe.

It was formerly used in medicine, and was supposed to cure agues, consumptions, &c., but has now, Class II.—Myriapoda.

This Class was proposed by Dr Leach in the Edinburgh Encyclopaedia, Vol. VII. and has since been distinctly established, and its characters more decidedly shown, in a paper published in the eleventh volume of the Transactions of the Linnean Society.

By Linné the animals composing this group were denominated Scolopendrae and Julli, and were arranged with apertous insects. His pupil J. C. Fabricius, in the Supplement to his Entomologia Systematica, placed them in a particular class named Mitosata, comprehending all the species, like Linné, under the generic appellations of Julius and Scolopendra. G. Cuvier, in his Tableau Élémentaire, arranged the Myriapoda with Insects, in which he was followed by A. M. C. Duméril, who has, however, adopted the new genera proposed by Latreille.

They were arranged in the older works of Latreille along with Insects; but in his last work he has placed them in a peculiar order of the class Arachnides, which he has denominated Myriapoda; and has divided them into two families, namely,

Fam. I. Chilognatha. Gen. 1. Glomeris. 2. Julius. 3. Polydesmus. 4. Pollyxenus.

Fam. II. Syngnatha. Gen. 5. Scutigera. 6. Scolopendra.

Lamarck arranged them with the Arachnides, into three genera, 1. Scolopenda; 2. Scutigera; 3 Julius; and in his last work, he had adopted a fourth genus, Pollyxenus.

Having given a slight sketch of what has been done by systematic writers, we may observe, that we differ from them merely in considering them as constituting a distinct class, and in disposing the species under some additional generic heads, which a minute examination of their structure has most fully warranted.

Classification.

All the Myriapoda have their head distinct from Myriapoda, the body, furnished with two antennae. Mandibles simple, incisive. All or most of the segments of the body furnished with two or four legs.

The nervous system is composed of a series of ganglia, one in each segment of the body; these ganglia are brought into communication with each other by a longitudinal bundle of nerves, or, as it is generally, but improperly, denominated, by a spinal marrow.

The two families established by Latreille, are adopted as Orders, and his names are retained.

Order I. Chilognatha. Maxilla none. Palpi obscure. Lip simple. (Antennae inserted on the upper margin of the head.)

Order II. Syngnatha. Maxillae two, distinct, with their bases united. Palpi, maxillary two, filiform; labial two terminated by a little hook.

Order I. Chilognatha.†

Fam. I. Glomeridea. Body contractile into a globe. Eyes distinct.

Gen. 1. Glomeris.

Fam. II. Julidea. Body not contractile into a globe. Eyes distinct.

Genus 2. Julius.

3. Chaspedosoma.

Fam. III. Polydesmidea. Eyes obsolete.

Genus 4. Polydesmus.

Family I. Glomeridea.

Gen. 1. Glomeris. Latr. Dumer. Leach.

Armadillo. Cuvier.

Antenna with the two first joints shortest, the sixth largest including the last, which is very small. Body elongate-ovate, convex above, arched beneath; first segment a little semicircular lamina, the second... Myriapoda larger than the others; the last semicircular and arched. Legs, sixteen pairs.

Sp. 1. Marginata. Black, the margins of the segments luteous or orange.

Oniscus marginatus.

Villers, Entom. iv. 187. tab. 11. fig. 15.

Glomeris bordé.

Latr. Hist. Nat. des Crust. et des Insect. vii. 66.

Oniscus marginatus.

Oliv. Encycl. Méth. Hist. Nat. vi. p. 24.

Julius limbatus.

Oliv. Encycl. Méth. Hist. Nat. vii. p. 414.

Oniscus zonatus.

Panz. Fn. Ins. Germ. Fascic. ix. f. 23.

Julius oniscoides.

Tocmson's Tracts, p. 151.

Stewart, Elem. Nat. Hist. ii. 307.

Glomeris marginata.

Latr. Gen. Crust. et Insect. i. 74.

Leach, Edinb. Encycl. vii. 407.

Trans. Linn. Soc. xi.

Plate XXII.

Inhabits Britain, France, and Germany, under stones; but has generally been considered by British Naturalists as a variety of Armadillo vulgaris.

Family II. Julidea.

Gen. 2. Julius of authors.

Body serpentiniform, cylindric. Antennae with the second joint longer than the third. Legs a great many.

The British species of this obscure genus may be found described in the eleventh volume of the Transactions of the Linnean Society. The following species, which is the most common, will best serve as an example of the genus.

Sp. 1. Sabulosus. Black-cinerous, with two reddish dorsal lines; last joint mucronated; legs luteous.

Julius sabulosus.

Linn. Syst. Nat. i. 1065.

Fabr. Ent. Syst. ii. 395.

Latr. Gen. Crust. et Insect. i. 76.

Leach, Edinb. Encycl. vii. 407.

Trans. Linn. Soc. xi.

Inhabits Europe, lurking beneath stones, especially in sandy places.

Gen. 3. Craspedosoma. Leach.

Body linear, depressed, the sides of the segments laterally prominent. Antennae towards their extremities somewhat thicker, the second joint shorter than the third.

This genus was discovered by the late R. Rawlins, Esq., one of the most promising naturalists of this country.

* Middle of the segments prominent.

Sp. 1. Raulinisii. Back fuscous-brown, with four lines of white spots; belly and legs reddish.

Craspedosoma Raulinisii.

Leach, Edinb. Encycl. vii. 407—434.

Trans. Linn. Soc. xi. 380.

Plate XXII.

Inhabits the neighbourhood of Edinburgh, where it occurs in some plenty under stones and amongst moss. It was first noticed by Mr Rawlins, the founder of the genus.

** Hinder angles of the segments produced.

Sp. 2. Polydesmoides. Body reddish-gray; belly Myriapoda pale; legs reddish, with their bases pale; produced angles of the body each furnished with a seta.

Julius polydesmoides. Montagu's MSS.

Craspedosoma polydesmoides.

Leach, Edinb. Encycl. vii. 407—434.

Trans. Linn. Soc. xi. 380.

Plate XXII.

Inhabits Devonshire, under stones. It is common all along the borders of Dartmoor, and on the southern coast.

Family III. Polydesmidea.

Gen. 4. Polydesmus. Latr. Dumér. Leach.

Antennae with the second joint scarcely longer than the first, and much shorter than the third. Body linear, the segments laterally compressed, margined. Eyes obsolete.

Sp. 1. Complanatus. Reddish cinereous, last segment of the body mucronated.

Julius complanatus.

Linn. Syst. Nat. i. 1065.

Fabr. Ent. Syst. ii. 395.

Polydesmus complanatus.

Latr. Gen. Crust. et Insect. i. 76.

Leach, Edinb. Encycl. vii. 408.

Trans. Linn. Soc. xi. 381.

Plate XXII.

Inhabits Europe, under stones.

Order II. Synognatha.

Fam. I. Cermatidea. Body with the segments each bearing four legs.

Genus 1. Cermatia.

Fam. II. Scolopendridae. Body with each segment bearing two legs: hinder legs distinctly longer than the others.

Stirps 1. Legs on each side fifteen.

Genus 2. Lithobius.

Stirps 2. Legs on each side twenty-one.

Genus 3. Scolopendra.

Cryptops.

Family III. Geophilidea. Body with each segment bearing two legs: hinder legs not distinctly longer than the others. Legs many, varying in number in the same species.

Genus 5. Geophilus.

Family I. Cermatidea.

Genus 1. Cermatia. Illiger, Leach.

Scutigera. Lam. Latr. Dumér. Leach.

Legs thirty.

Sp. 1. Coleopterata. Body reddish-yellowish, with longitudinal lines and bars on the legs of blue black.

Scolopendra coleopterata.

Linn. Syst. Nat. i. 1062.

Fabr. Ent. Syst. ii. 389.

Julius araneoides. Pall. Spec. Zool. Fas. ix. t. 4. f. 16.

Scutigera araneoides.

Latr. Gen. Crust. et Insect. i. 77.

Scutigera coleopterata.

Leach, Edinb. Encycl. vii. 408.

Inhabits houses in the south of Europe. It is common also in Africa.

Family II. Scolopendridae.

Gen. 2. Lithobius. Leach.

Antennae conic-setaceous (joints about forty-five), conic-setaceous, the two first joints largest. Under