page 98 note * Herschel: Discourse on the Study of Natural Philosophy; Herbert Spencer: Essays, Scientific, Political, and Speculative, Vols. 1 and 3; Whewell: Novum Organon Renovatuni; Jevons: The Principles of Science; Mill: A System of Logic, Vols. 1 and 2; and Bain: Logic: Part II, on Induction.

page 99 note * Employing this term as distinctive from the technical phase of the “Force of Mortality.”

page 100 note * Spencer: The Classification of the Sciences: Essays, VOL. 3. Bain: Logic: Part I: Deduction: Appendix A.

^{page 100 note †} Whewell: “Bridgewater” Treatise on Astronomy and General Physics; Introduction, cap. 2. Spencer: The Study of Sociology, cap. 2.

page 101 note * If indeed Science should not hereafter regard the Universe as simply Matter under changing forms or purely Force in varying modes.

page 102 note * The statistical information above presented, expresses the experience of one of our principal London Hospitals, and may be accepted, I am assured, as typical of that prevailing in the Metropolitan Hospitals generally.

The subject is so impressive in relation to our data, that a brief reference to the disease of Pulmonary Consumption should not be omitted. The view formerly adopted regarded Phthisis Pulmonalis as essentially originating within the system, though various elements contributed a predisposing force; but the discovery by Dr. Koch and others of the existence of vegetable organisms or bacilli in tubercular growths has materially modified the ancient teaching. With the doctrine of the dependence of Tuberculosis upon vital organisms, which must of necessity be introduced within the system from external sources, attention to exterior agencies may now be deemed to be the effective instrument for protection or mitigation. We are thus confronted with the two factors of (i) a possible constitutional predisposition, or (to speak figuratively) a possible provision of appropriate pabulum for these pathogenic organisms, and (ii) the possible prevention, even where such a diathesis exists, of the intrusion of bacilli by suitable conditions of isolation and physical environment.

^{page 102 note †} Whewell: Cambridge Philosophical Transactions for 1829, 1831, and 1850.

^{page 102 note ‡} Jevons: The Theory of Political Economy.

page 103 note * Cairns: The Character and Logical Method of Political Economy.

page 104 note * A Treatise on the Valuation of Annuities and Assurances; Vol. 2; Appendix No. 5.

^{page 104 note †} The Study of Sociology, cap. 5.

^{page 104 note ‡} Whewell: Novum Organon Renovatum, cap. 4.

^{page 104 note §} Mill: A System of Logic, Vol. I, Lib. 3, cap. 2.

page 105 note * Philosophical Transactions of the Royal Society: reprinted in the Journal of the Institute of Actuaries, Vol. xyiii, pp. 251 and 262.

^{page 105 note †} Smart: Tables of Interest and Annuities.

^{page 105 note ‡} In connexion with the employment of these observations for Mortality deductions.

^{page 105 note §} Journal of the Institute of Actuaries: Vol. ii, pp. 121, 222: Vol. in, p. 93. [Vide Note on p. 121.]

page 106 note * This consideration was especially significant in Halley's time, when the population of London, as we may infer from Macaulay's History of England, only approximated to about 500,000 persons.

page 107 note * Journal of the Institute of Actuaries, Vol. xiv, p. 247.

^{page 107 note †} Journal of the Institute, Vol. i, p. 179; Vol. xv, p. 328; Vol. xx, p. 95; Vol. xxi, p. 229; Vol. xxii, p. 391.

In the Tabulation of the numbers collected, and the deduction of the ratios involved, I might include the Conception of the appropriate mode of operation. The Tables of Halley, Simpson, and Price were based upon the deaths, and the accurate method was first fully pursued in Milne's investigations where the deaths occurring at each age are brought into relation with the corresponding numbers living.

page 108 note * Hypotheses or Suppositions are the mental conceptions which, under the tendency of the mind towards generalisation and unity, are provisionally “placed beneath” (as the term etymologically signifies), the apparently disconnected facts as their rational support and explanation. And although scientific language is not precisely determinate upon the point, we may legitimately affirm that an hypothesis, which has received adequate confirmation by recognised scientific tests, may, at that supreme moment, be designated a Theory, or an original fact of Nature which the Speculator or Ideal Spectator (for this etymological implication the term “Theory” involves) would, if gifted with commanding power of mental vision, directly observe as an integral constituent of the Physical Scheme.

^{page 108 note †} Mill: A System of Logic, Vol. 2, Lib. 3, cap. 14.

^{page 108 note ‡} Bacon: Novum Organum: Lib. 1, Aph. 26.

^{page 108 note §} Newton: Principia, Lib. 3.

page 110 note * De Moivre: Annuities upon Lives.

^{page 110 note †} Gompertz: Philosophical Transactions of the Royal Society.

page 111 note * Journal of the Institute of Actuaries: Vol. IV, p. 199.

^{page 111 note †} Journal of the Institute: Vol. vii, p. 121; Vol. viii, p. 181; Vol. xiii, p. 325; Vol. xv, p. 389.

page 112 note * Journal of the Institute of Actuaries: Vol. xvi, p. 329.

^{page 112 note †} Journal of the Institute: Vol. viii, p. 301; Vol. xxviii, pp. 152,185, and 316.

page 113 note * Reference might also be made to an investigation (Journal of the Institute of Actuaries: Vol. xvii, p. 56) instituted into the law supposed to be exhibited in the ages at which Assurances are effected. Even if this uniformity were established, it would prove to be simply of an empirical character implying no adequate ground for extension beyond the individual or composite experience in which it might be observed.

^{page 113 note †} By Professor Keeler.

^{page 113 note ‡} Journal of the Institute of Actuaries: Vol. viii, p. 181.

^{page 113 note §} A Treatise on the Valuation of Annuities and Assurances: Introduction, p. 51.

^{page 113 note ∥} Journal of the Institute: Vol. ix, p. 305.

page 114 note * Novum Organum: Lib. ii: Aph, 39.

^{page 114 note †} It is curious to notice that the interpretation of this Table has not proved generally evident. Montucla conceived that the number (1000) placed against “age cur^t.1” expressed the births: Daniel Bernouilli understood that the number of infants born was not furnished, and that the 1000 were supposed to attain the age of 1: he accordingly estimated the radix of the Table to be 1300; while Farren considered that the 1000 represented the number of children who were aged one year.

^{page 114 note ‡} The Doctrine of Annuities and Reversions: containing a Table deduced from the London Bills of Mortality.

^{page 114 note §} Journal of the Institute of Actuaries; Vol. i, p. 15: “Calculations deduced from First Principles.”

page 115 note * Journal of the Institute of Actuaries: Vol. i, p. 2.

^{page 115 note †} Journal of the Institute, Vol. i, p. 1; Vol. iv, p. 185.

^{page 115 note ‡} Tables of Life Contingencies.

^{page 115 note §} Journal of the Institute: Vol. xv, p. 96.

page 116 note * Journal of the Institute of Actuaries: Vol. xiv, p. 249.

^{page 116 note †} Whewell: NOT. Org. Renov.: Lib. 3, cap. vi.

Balfour Stewart: A Treatise on Heat: Lib. ii, cap. 4.

page 117 note * J. Finlaison: On the Evidence and Elementary Facts on which the Tables of Life Annuities are founded.

^{page 117 note †} Journal of the Institute of Actuaries: Vol. xxii, p. 24.

^{page 117 note ‡} Journal of the Institute: Vol. xv, p. 389.

page 118 note * Journal of the Institute of Actuaries: Vol. xxvi, p. 77.

^{page 118 note †} Herschel: Investigation of the Orbits of Doable Stars;

Herschel: Meteorology, § 29.

Whewell: Researches on the Tides.

^{page 118 note ‡} Farren, e.g.: The Chances of Premature Death and the Value of Selection.

page 119 note * Dr. Price, e.g. under a preconception, arbitrarily altered the decrements of the Northampton table between the ages of 20 and 30. (Journal of the Institute of Actuaries: Vol. v, p. 284.)

page 120 note * Ball: A History of Mathematics.

^{page 120 note †} Carnot: Reflections on the Metaphysical Principles of the Infinitesimal Analysis.

page 121 note * Recent investigations into the early history of Life Contingencies in Holland have been fertile of interesting discoveries. We now learn (i) that when de Witt's Treatise was presented to the Assembly of the States-General of Holland and West Friesland in 1671, it was at once ordered to be officially printed. This was done; and the document was circulated among the Members of the States; probably about 100 in number; (ii) Extracts from the Resolutions of the States of Holland and West Friesland were regularly printed officially shortly after the Resolutions had been passed. In the Volume containing Extracts, dated December 1670, de Witt's Report again appears under date of the 30th of July 1671. This Volume I have had an opportunity of inspecting, and copies are exceedingly rare; and (iii) the manuscript of the Treatise still remains in the Archives of State at the Hague.

page 122 note * Annuities on lives: a_x=vp_x (1 +a_x+1).

The distinguished mathematician, Euler,—as we learn from Todhunter's History of the Mathematical Theory of Probability,—investigated the problem in 1760 (published in 1767), and showed that the value at any age provided a mode of immediately determining the value at the succeeding age. This conclusion was apparently deduced quite independently of the enquiries of his predecessors, and the precise form of the modern expression is, of course, at once obtained by a simple algebraical transformation.

^{page 122 note †} Ball: A History of Mathematics: Cap. v.

^{page 122 note ‡} Simpson: The Doctrine of Annuities and Reversions.

^{page 122 note §} Journal of the Institute of Actuaries: Vol. xviii, p. 305.

^{page 122 note ∥} Journal of the Institute: Vol. xi, p. 301; Vol. xv, p. 95.

^{page 122 note ¶} Journal of the Institute: Vol. xxiv, p. 95: Vol. xxvi, p. 276 Vol. xxvii, p. 122.

^{page 122 note **} Journal of the Institute: Vol. i, p. 96.

^{page 122 note ††} The Doctrine of Annuities and Reversions.

page 123 note * Cheyne: Treatise on the Planetary Theory. Godfray: Treatise on the Lunar Theory.

^{page 123 note †} Treatise on Natural Philosophy. Vol. ii, cap. 5.

page 124 note * An Experimental Enquiry into the Nature of Heat: Note iv.

page 125 note * Aphorisms on the Language of Science: Nov. Org. Renov.

^{page 125 note †} A System of Logic: Vol. ii., Lib. iv., cap. 4 to 6.

page 126 note * Journal of the Institute of Actuaries: Vol. vii, p. 136.

^{page 126 note †} Treatise on Annuities: 4th edition.

^{page 126 note ‡} Mathematical Repository.

^{page 126 note §} Treatise on Annuities: cap. iv., § 5.

^{page 126 note ∥} Journal of the Institute: Vol. xx, p. 435.

^{page 126 note ¶} Journal of the Institute: Vol. x, p. 147.

^{page 126 note **} Journal of the Institute: Vol. viii, P. 127; Vol. x, P. 181: Vol. xii, P. 185; and Vol. xxi, p. 406.

page 128 note * As the Science of Numbers is the basis of every mode of Actuarial work, early speculations and applications constituted a rich armoury of weapons with which, as observations became more complete and specialised, succeeding enquirers could successfully attack the increasing complexity of questions which the progress of our science produced. The investigations of Daniel Bernouilli in 1760, of D'Alembert in 1761, and of Laplace in 1812, into the mortality due to smallpox, are instructive examples of mathematical skill applied to limited data.

^{page 128 note †} Gompertz: Memoir of 1820.

page 130 note * Bacon: De Angmentis Scientiaram: Lib, v, cap. 3.