Chapter XI: Appendix (10)

So likewise for the _Odds_, that any Person does not die before he attain any proposed _Age_: Take the _number_ of the remaining Persons of the _Age_ proposed, and divide it by the difference between it and the number of those of the _Age_ of the Party proposed; and that shews the _Odds_ there is between the Chances of the Party's living or dying. As for Instance; What is the _Odds_ that a Man of 40 lives 7 Years: Take the number of Persons of 47 Years, which in the Table is 377, and subtract it from the number of Persons of 40 Years, which is 445, and the _difference_ is 68: Which shews that the _Persons dying_ in that 7 Years, are 68, and that it is 377 to 68, or 5½ to 1, that a Man of 40 does live 7 Years. And the like for any other _number_ of _Years_.

_Use_ III. But if it be enquired at what number of _Years_, it is an even Lay that a Person of any _Age_ shall die, this Table readily performs it; For if the _number_ of Persons _living_ of the _Age_ proposed, be _halfed_, it will be found by the _Table_ at what Year the said _Number_ is reduced to half by _Mortality_; and that is the _Age_, to which it is an even Wager, that a Person of the _Age_ proposed shall arrive before he _die_. As for Instance; A Person of 30 Years of _Age_ is proposed, the number of that _Age_ is 531, the half thereof is 265, which number I find to be between 57 and 58 Years; so that a Man of 30 may reasonably expect to live between 27 and 28 Years.

_Use_ IV. By what has been said, the _Price_ of _Insurance_ upon Lives ought to be regulated, and the difference is discovered between the _Price_ of insuring the _Life_ of a _Man_ of 20 and 50. For Example; It being 100 to 1, that a Man of 20 dies not in a Year, and but 38 to 1, for a Man of 50 Years of Age.

_Use_ V. On this depends the Valuation of _Annuities_ upon _Lives_; for it is plain, that the _Purchaser_ ought to pay for only such a part of the Value of the _Annuity_, as he has Chances that he is living; and this ought to be computed yearly, and the Sum of all those yearly Values being added together, will amount to the Value of the _Annuity_ for the _Life_ of the Person proposed. Now the present Value of Money payable after a Term of Years, at any given Rate of Interest, either may be had from Tables already computed; or almost as compendiously, by the Table of Logarithms: For the Arithmetical Complement of the Logarithm of Unity, and its yearly Interest, (that is, of 1,06 for Six _per Cent._ being 9,974694.) being multiplied by the number of Years proposed, gives the present Value of One Pound payable after the end of so many Years. Then by the foregoing Proposition, it will be as the number of Persons living after that Term of Years, to the number dead; so are the Odds that any one Person is alive or dead. And by consequence, as the Sum of both, or the number of Persons living of the _Age_ first proposed, to the number remaining after so many Years, (both given by the Table) so the present Value of the yearly Sum payable after the Term proposed, to the Sum which ought to be paid for the Chance the Person has to enjoy such an _Annuity_ after so many Years. And this being repeated for every Year of the Person's Life, the Sum of all the present Values of those Chances is the true Value of the Annuity. This will without doubt appear to be a most laborious Calculation; but it being one of the principal Uses of this Speculation, and having found some _Compendia_ for the Work, I took the pains to compute the following Table, being the short Result of a not ordinary number of Arithmetical Operations: It shews the Value of Annuities for every Fifth Year of Age, to the Seventieth, as follows.

+------+------------+------+------------+------+------------+ | Age. | Years Pur. | Age. | Years Pur. | Age. | Years Pur. | +------+------------+------+------------+------+------------+ | 1 | 10,28 | 25 | 12,27 | 50 | 9,21 | | 5 | 13,40 | 30 | 11,72 | 55 | 8,51 | | 10 | 13,44 | 35 | 11,12 | 60 | 7,60 | | 15 | 13,33 | 40 | 10,57 | 65 | 6,54 | | 20 | 12,78 | 45 | 9,91 | 70 | 5,32 | +------+------------+------+------------+------+------------+

This shews the great Advantage of putting Money into the present _Fund_ lately granted to Their Majesties, giving 14 _per Cent. per Annum_, or at the Rate of 7 Years Purchase for a Life; when young Lives, at the usual Rate of Interest, are worth above 13 Years Purchase. It shews likewise the Advantage of young Lives over those in Years; a Life of Ten Years being almost worth 13½ Years Purchase, whereas one of 36 is worth but 11.

_Use_ VI. Two Lives are likewise valuable by the same Rule; for the number of Chances of each single Life, found in the Table, being multiplied together, become the Chances of the Two Lives. And after any certain Term of Years, the Product of the two remaining Sums is the Chances that both the Persons are living. The Product of the two Differences, being the numbers of the Dead of both Ages, are the Chances that both the Persons are dead. And the two Products of the remaining Sums of the one Age multiplied by those dead of the other, shew the Chances that there are, that each Party survives the other: Whence is derived the Rule to estimate the Value of the Remainder of one Life after another. Now as the Product of the Two Numbers in the Table for the Two Ages proposed, is to the difference between that Product, and the Product of the two numbers of Persons deceased in any space of time; so is the Value of a Sum of Money to be paid after so much time, to the Value thereof under the Contingency of Mortality. And as the aforesaid Product of the two Numbers answering to the Ages proposed, to the Product of the Deceased of one Age multiplied by those remaining alive of the other; so the Value of a Sum of Money to be paid after any time proposed, to the Value of the Chances, that the one Party has that he survives the other, whose number of Deceased you made use of, in the second Term of the Proportion. This perhaps may be better understood, by putting _N_ for the number of the younger Age, and _n_ for that of the Elder; _Y_, _y_ the Deceased of both Ages respectively, and _R_, _r_ for the Remainders; and _R + Y_ = _N_, and _r + y_ = _n_. Then shall _Nn_ be the whole Number of Chances; _Nn - Yy_ be the Chances that one of the two Persons is living, _Yy_ the Chances that they are both dead; _Ry_ the Chances that the elder Person is dead, and the younger living; and _rY_ the Chances, that the elder is living, and the younger dead. Thus two Persons of 18 and 35 are proposed, and after 8 Years these Chances are required. The Numbers for 18 and 35, are 610 and 490; and there are 50 of the First Age dead in 8 Years, and 73 of the Elder Age. There are in all 610 × 490, or 298900 Chances; of these there are 50 × 73, or 3650, that they are both dead. And as 298900, to 298900 - 3650, or 295250: So is the present Value of a Sum of Money to be paid after 8 Years, to the present Value of a Sum to be paid, if either of the two live. And as 560 × 73, so are the Chances that the Elder is dead, leaving the Younger; and as 417 × 50, so are the Chances that the Younger is dead, leaving the Elder. Wherefore as 610 × 490 to 560 × 73, so is the present Value of a Sum to be paid at 8 Years end, to the Sum to be paid for the Chance of the Younger's Survivance; and as 610 × 490 to 417 × 50, so is the same present Value to the Sum to be paid for the Chance of the Elder's Survivance.

This possibly may be yet better explained, by expounding these Products by Rectangular Parallelograms, as in _Fig. 7._ wherein _AB_ or _CD_ represents the number of Persons of the younger Age, and _DE_, _BH_ those remaining alive after a certain Term of Years; whence _CE_ will answer the number of those dead in that time: So _AC_, _BD_ may represent the number of the elder Age; _AF_, _BI_ the Survivors after the same Term; and _CF_, _DI_, those of that Age that are dead at that time. Then shall the whole Parallelogram _ABCD_ be _Nn_, or the Product of the two Numbers of Persons, representing such a number of Persons of the two Ages given; and by what was said before, after the Term proposed, the Rectangle _HD_ shall be as the number of Persons of the younger Age that survive, and the Rectangle _AE_ as the number of those that die. So likewise the Rectangles _AI_, _FD_ shall be as the Numbers, living and dead, of the other Age. Hence the Rectangle _HI_ shall be as an equal number of both Ages surviving. The Rectangle _FE_ being the Product of the Deceased, or _Yy_, an equal number of both dead. The Rectangle _GD_ or _Ry_, a number living of the younger Age, and dead of the elder: And the Rectangle _AG_ or _rY_ a number living of the elder Age, but dead of the younger. This being understood, it is obvious, that as the whole Rectangle _AD_ or _Nn_ is to the _Gnomon FABDEG_ or _Nn - Yy_, so is the whole number of Persons or Chances, to the number of Chances that one of the two Persons is living: And as _AD_ or _Nn_ is to _FE_ or _Yy_, so are all the Chances, to the Chances that both are dead; whereby may be computed the Value of the Reversion after both Lives. And as _AD_ to _GD_ or _Ry_, so the whole number of Chances, to the Chances that the younger is living, and the other dead; whereby may be cast up what Value ought to be paid for the Reversion of one Life after another, as in the Case of providing for Clergy-men's Widows, and others, by such Reversions. And as _AD_ to _AG_, or _rY_, so are all the Chances, to those that the elder survives the younger. I have been the more particular, and perhaps tedious, in this Matter, because it is the Key to the Case of Three Lives, which of it self would not have been so easie to comprehend.

VII. If Three Lives are proposed, to find the Value of an Annuity during the continuance of any of those three Lives; the Rule is, _As the Product of the continual Multiplication of the Three Numbers, in the Table, answering to the Ages proposed, is to the difference of that Product, and of the Product of the Three Numbers of the Deceased of those Ages, in any given Term of Years: So is the present Value of a Sum of Money, to be paid certainly after so many Years, to the present Value of the same Sum to be paid, provided one of those Three Persons be living at the Expiration of that Term._ Which Proportion being yearly repeated, the Sum of all those present Values will be the Value of an Annuity granted for three such Lives. But to explain this, together with all the Cases of Survivance in Three Lives: Let _N_ be the Number in the Table for the younger Age, _n_ for the second, and ν for the elder Age; let _Y_ be those dead of the younger Age in the Term proposed, _y_ those dead of the second Age, and υ those of the elder Age; and let _R_ be the Remainder of the younger Age, _r_ that of the middle Age, and ρ the Remainder of the elder Age. Then shall _R + Y_ be equal to _N_, _r + y_ to _n_, and ρ + υ to ν, and the continual Product of the three Numbers _N_, _n_, ν, shall be equal to the continual Product of _R + Y × r + y × ρ + υ_, which being the whole Number of Chances for three Lives, is compounded of the eight Products following. (1) _Rrρ_, which is the Number of Chances that all three of the Persons are living. (2) _rρY_, which is the Number of Chances that the two elder Persons are living, and the younger dead. (3) _Rρy_ the Number of Chances that the middle Age is dead, and the younger and elder living. (4) _Rrυ_ being the Chances that the two younger are living, and the elder dead. (5) _ρYy_ the Chances that the two younger are dead, and the elder living. (6) _rYυ_ the Chances that the younger and elder are dead, and the middle Age living. (7) _Ryυ_, which are the Chances that the younger is living, and the two other dead. And Lastly and Eighthly, _Yyυ_, which are the Chances that all three are dead. Which latter subtracted from the whole Number of Chances _Nnν_, leaves _Nnν - Yyυ_ the Sum of all the other seven Products; in all of which one or more of the three Persons are surviving.

To make this yet more evident, I have added _Fig. 8._ wherein these eight several Products are at one view exhibited. Let the rectangled Parallelepipedon _ABCDEFGH_ be constituted of the sides _AB_, _GH_, _&c._ proportional to _N_ the Number of the younger Age; _AC_, _BD_, _&c._ proportional to _n_; and _AG_, _CE_, _&c._ proportional to the Number of the elder, or ν. And the whole Parallelepipedon shall be as the Product _Nnν_, or our whole Number of Chances. Let _BP_ be as _R_, and _AP_ as _Y_; let _CL_ be as _r_, and _Ln_ as _y_; and _GN_ as ρ, and _NA_ as υ; and let the Plain _PRea_ be made parallel to the Plain _ACGE_; the Plain _NVbY_ parallel to _ABCD_; and the Plain _LXTQ_ parallel to the Plain _ABGH_. And our first Product _Rrρ_ shall be as the Solid _STWIFZeb_. The Second, or _rρY_ will be as the Solid _EYZeQSMI_. The Third, _Rρy_, as the Solid _RHOVWIST_. And the Fourth, _Rrυ_, as the Solid _ZabDWXIK_. Fifthly, _ρYy_, as the Solid _GQRSIMNO_. Sixthly, _rYυ_, as _IKLMGYZA_. Seventhly, _Ryυ_, as the Solid _IKPOBXVW_. And Lastly, _AIKLMNOP_ will be as the Product of the 3 Numbers of Persons dead, or _Yyυ_. I shall not apply this in all the Cases thereof, for brevity sake; only to shew in one how all the rest may be performed, let it be demanded what is the Value of the Reversion of the younger Life after the two elder proposed. The proportion is as the whole Number of Chances, or _Nnν_ to the Product _Ryυ_; so is the certain present Value of the Sum payable after any Term proposed, to the Value due to such Chances as the younger Person has to bury both the elder, by the Term proposed; which therefore he is to pay for. Here it is to be noted, that the first Term of all these Proportions is the same throughout, _viz._ _Nnν_. The second changing yearly according to the Decrease of _R_, _r_, _ρ_, and Increase of _Y_, _y_, _υ_. And the third are successively the present Values of Money payable after one, two, three, _&c._ years, according to the Rate of Interest agreed on. These Numbers, which are in all Cases of Annuities of necessary Use, I have put into the following Table, they being Decimal Values of one Pound payable after the Number of Years in the Margent, at the Rate of 6 _per Cent._

+--------+-------------+--------+-------------+--------+-------------+ | Years. | Pres. Value | Years. | Pres. Value | Years. | Pres. Value | | | of 1 _l._ | | of 1 _l._ | | of 1 _l._ | +--------+-------------+--------+-------------+--------+-------------+ | 1 | 0,9434 | 19 | 0,3305 | 37 | 0,1158 | | 2 | 0,8900 | 20 | 0,3118 | 38 | 0,1092 | | 3 | 0,8396 | 21 | 0,2941 | 39 | 0,1031 | | 4 | 0,7921 | 22 | 0,2775 | 40 | 0,0972 | | 5 | 0,7473 | 23 | 0,2618 | 45 | 0,0726 | | 6 | 0,7050 | 24 | 0,2470 | 50 | 0,0543 | +--------+-------------+--------+-------------+--------+-------------+ | 7 | 0,6650 | 25 | 0,2330 | 55 | 0,0406 | | 8 | 0,6274 | 26 | 0,2198 | 60 | 0,0303 | | 9 | 0,5919 | 27 | 0,2074 | 65 | 0,0227 | | 10 | 0,5584 | 28 | 0,1956 | 70 | 0,0169 | | 11 | 0,5268 | 29 | 0,1845 | 75 | 0,0126 | | 12 | 0,4970 | 30 | 0,1741 | 80 | 0,0094 | +--------+-------------+--------+-------------+--------+-------------+ | 13 | 0,4688 | 31 | 0,1643 | 85 | 0,0071 | | 14 | 0,4423 | 32 | 0,1550 | 90 | 0,0053 | | 15 | 0,4173 | 33 | 0,1462 | 95 | 0,0039 | | 16 | 0,3936 | 34 | 0,1379 | 100 | 0,0029 | | 17 | 0,3714 | 35 | 0,1301 | | | | 18 | 0,3503 | 36 | 0,1227 | | | +--------+-------------+--------+-------------+--------+-------------+

It were needless to advertise, that the great trouble of working so many Proportions will be very much alleviated by using Logarithms; and that instead of using _Nnν - Yyυ_ for the second Term of the Proportion in finding the Value of Three Lives, it may suffice to use only _Yyυ_, and then deducting the fourth Term so found out of the third, the Remainder shall be the present Value sought; or all these fourth Terms being added together, and deducted out of the Value of the certain Annuity for so many Years, will leave the Value of the contingent Annuity upon the Chance of Mortality of all those Three Lives. For Example; Let there be Three Lives of 10, 30, and 40 Years of Age proposed, and the Proportions will be thus;

As 661 in 531 in 445 or 156190995, or _Nnν_ to 8 in 8 in 9, or 576, or _Yyυ_ for the first Year, so 0,9434. to 0,00000348.

To 15 in 16 in 18, or 4320, for the second Year, so 0,8900. to 0,00002462.

To 21 in 24 in 28, or 14112 for the third Year, so 0,8396. to 0,00008128.

To 27 in 32 in 38, for the fourth Year, so 0,7921. to 0,00016650.

To 33 in 41 in 48, for the fifth Year, so 0,7473. to 0,00031071.

To 39 in 50 in 58, for the sixth Year, so 0,7050. to 0,00051051.

And so forth to the 60th Year, when we suppose the elder Life of Forty certainly to be expired; from whence till Seventy we must compute for the First and Second only, and from thence to Ninety for the single youngest Life. Then the Sum Total of all these Fourth Proportionals being taken out of the Value of a certain Annuity for 90 Years, being 16,58 Years Purchase, shall leave the just Value to be paid for an Annuity during the whole Term of the Lives of Three Persons of the Ages proposed. And note, that it will not be necessary to compute for every Year singly; but that in most Cases every 4th or 5th Year may suffice, interpoling for the intermediate Years _seceundum artem_.

It may be objected, that the different _Salubrity_ of Places does hinder this Proposal from being _universal_; nor can it be denied. But by the Number that die, being 1174. _per Annum_ in 34000, it does appear that about a 30th part die yearly, as Sir _William Petty_ has computed for _London_; and the Number that die in Infancy, is a good Argument that the Air is but indifferently salubrious. So that by what I can learn, there cannot perhaps be one better Place proposed for a Standard. At least 'tis desired, that in Imitation hereof the Curious in other Cities would attempt something of the same Nature, than which nothing perhaps can be more useful.

Were this _Calculus_ founded on the Experience of a very great number of Years, it would be very well worth the while to think of Methods for facilitating the Computation of the Value of two, three, or more Lives; which, as proposed in my former, seems (as I am inform'd) a Work of too much Difficulty for the ordinary Arithmetician to undertake.

I have sought, If it were possible, to find a Theorem that might be more concise than the Rules there laid down, but in vain; for all that can be done to expedite it, is, by Tables of Logarithms ready computed, to exhibit the _Rationes_ of _N_ to _Y_ in each single Life, for every third, fourth, or fifth Year of Age, as occasion shall require; and these Logarithms being added to the Logarithms of the present Value of Money payable after so many Years, will give a Series of Numbers, the Sum of which will shew the Value of the Annuity sought. However, for each Number of this Series, two Logarithms for a single Life, three for two Lives, and four for three Lives, must necessarily be added together. If you think the Matter, under the Uncertainties I have mentioned, to deserve it, I shall shortly give you such a Table of Logarithms, as I speak of, and an Example or two of the use thereof: But by Vulgar Arithmetick, the Labour of these Numbers were immense; and nothing will more recommend the useful Invention of Logarithms to all Lovers of Numbers, than the advantage of Dispatch in this and such like Computations.

Besides the Uses mentioned, it may perhaps not be an unacceptable thing to infer from the same Tables, how unjustly we repine at the shortness of our Lives, and think our selves wronged if we attain not old Age; whereas it appears hereby, that the one half of those that are born are dead in Seventeen Years time, 1238 being in that time reduced to 616. So that instead of murmuring at what we call an untimely Death, we ought with Patience and Unconcern to submit to that Dissolution which is the necessary Condition of our perishable Materials, and of our nice and frail Structure and Composition: And to account it as a Blessing that we have survived, perhaps by many Years, that Period of Life, whereat the one half of the whole Race of Mankind does not arrive.

A second Observation I make upon the said Table, is that the Growth and Increase of Mankind is not so much stinted by any thing in the Nature of the _Species_, as it is from the cautious difficulty most People make to adventure on the State of _Marriage_, from the Prospect of the Trouble and Charge of providing for a Family. Nor are the poorer sort of People herein to be blamed, since their difficulty of subsisting is occasion'd by the unequal Distribution of Possessions, all being necessarily fed from the Earth, of which yet so few are Masters. So that besides themselves and Families, they are yet to work for those who own the Ground that feeds them: And of such does by very much the greater part of Mankind consist; otherwise it is plain, that there might well be four times as many Births as we now find. For by Computation from the Table, I find that there are nearly 15000 Persons above 16, and under 45, of which at least 7000 are Women capable to bear Children. Of these notwithstanding there are but 1238 born yearly, which is but little more than a sixth part: So that about one in six of these Women do breed yearly; whereas were they all married, it would not appear strange or unlikely, that four of six should bring a Child every Year. The Political Consequences hereof I shall not insist on; only the Strength and Glory of a King being in the multitude of his Subjects, I shall only hint, that above all things, Celibacy ought to be discouraged, as, by extraordinary Taxing and Military Service: And those who have numerous Families of Children to be countenanced and encouraged by such Laws as the _Jus trium Liberorum_ among the _Romans_. But especially, by an effectual Care to provide for the Subsistence of the Poor, by finding them Employments, whereby they may earn their Bread, without being chargeable to the Publick.

_A Discourse concerning _Gravity_, and its Properties, wherein the
Descent of _Heavy Bodies_, and the Motion of _Projects_ is briefly,
but fully handled: Together with the _Solution_ of a _Problem_ of
great Use in _Gunnery_. By _E. Halley_._

Nature, amidst the great Variety of _Problems_, wherewith She exercises the Wits of Philosophical Men, scarce affords any one wherein the Effect is more visible, and the Cause more concealed, than in those of the _Phænomena_ of _Gravity_. Before we can go alone, we must learn to defend our selves from the Violence of its Impulse, by not trusting the _Center_ of _Gravity_ of our Bodies beyond our reach; and yet the acutest Philosophers, and the subtilest Enquirers into the Original of this Motion, have been so far from satisfying their Readers, that they themselves seem little to have understood the Consequences of their own _Hypotheses_.

_Des Cartes_ his Notion, I must needs confess to be to me incomprehensible, while he will have the Particles of his _Cœlestial Matter_, by being reflected on the Surface of the _Earth_, and so ascending therefrom, to drive down into their Places those _Terrestrial Bodies_ they find above them: This is, as near as I can gather, the Scope of the 20, 21, 22, and 23 _Sections_ of the last Book of his _Principia Philosophiæ_; yet neither he, nor any of his Followers, can shew how a Body suspended in _Libero Æthere_, shall be carried downwards by a continual Impulse tending upwards, and acting upon all its Parts equally: And besides the Obscurity wherewith he expresses himself, particularly, _Sect. 23._ does sufficiently argue according to his own Rules, the confused _Idea_ he had of the thing he wrote.

Others, and among them Dr. _Vossius_, assert the Cause of the _Descent_ of _heavy Bodies_, to be the _Diurnal Rotation_ of the _Earth_ upon its _Axis_, without considering, that according to the Doctrine of Motion fortified with Demonstration, all Bodies moved _in Circulo_, would recede from the Center of their Motion; whereby the contrary to _Gravity_ would follow, and all loose Bodies would be cast into the Air in a _Tangent_ to the _Parallel_ of _Latitude_, without the intervention of some other Principle to keep them fast, such as is that of _Gravity_. Besides, the Effect of this Principle is throughout the whole Surface of the Globe found nearly equal; and certain Experiments have proved it rather less near the _Æquinoctial_, than towards the _Poles_; which could not be by any means, if the _Diurnal Rotation_ of the _Earth_ upon its _Axis_ were the Cause of _Gravity_; for where the Motion was swiftest, the Effect would be most considerable.

Others assign the Pressure of the _Atmosphere_, to be the Cause of this Tendency towards the Center of the Earth; but unhappily they have mistaken the Cause for the Effect; it being from undoubted Principles plain, that the _Atmosphere_ has no other Pressure but what it derives from its _Gravity_; and that the Weight of the upper Parts of the _Air_, pressing on the lower Parts thereof, do so far bend the Springs of that _Elastick_ Body, as to give it a Force equal to the Weight that compress'd it, having of it self no force at all: And supposing it had, it will be very hard to explain the _Modus_, how that Pressure should occasion the Descent of a Body circumscribed by it, and pressed equally above and below, without some other Force to draw, or thrust it downwards. But to demonstrate the contrary of this Opinion, an _Experiment_ was long since shewn before the _Royal Society_, whereby it appeared, that the _Atmosphere_ was so far from being the Cause of _Gravity_, that the Effects thereof were much more vigorous, where the Pressure of the _Atmosphere_ was taken off; for a long _Glass-Receiver_ having a light Down-feather included, being evacuated of Air, the Feather, which in the Air would hardly sink, did _in vacuo_ descend with nearly the same _Velocity_, as if it had been a Stone.

Some think to illustrate this Descent of Heavy Bodies, by comparing it with the Vertue of the _Loadstone_; but setting aside the difference there is in the manner of their Attractions, the _Loadstone_ drawing only in and about its Poles, and the Earth near equally in all Parts of its Surface, this Comparison avails no more than to explain _ignotum per æque ignotum_.

Others assign a certain _Sympæthetical Attraction_ between the Earth and its Parts, whereby they have, as it were, a desire to be united, to be the Cause we enquire after: But this is so far from explaining the _Modus_, that it is little more, than to tell us in other Terms, that Heavy Bodies descend, because they descend.

This, I say, not that I can pretend to substitute any Solution of this Important Philosophical Problem, that shall more happily explicate the Appearances of Gravity; only it may be serviceable to those with whom the Credit of great Authors sways much, and who too readily assent _in Verba Magistri_, to let them see that their Books are not always infallible: Besides, the detection of Errors is the first and surest Step towards the discovery of Truth.

Though the efficient Cause of _Gravity_ be so obscure, yet the final Cause thereof is clear enough; for it is by this single _Principle_, that the _Earth_ and all the _Cœlestial Bodies_ are kept from _Dissolution_; the least of their _Particles_ not being suffer'd to recede far from their _Surfaces_, without being immediately brought down again by Virtue of this _Natural Tendency_; which, for their Preservation, the Infinite Wisdom of their _Creator_ has ordained to be towards each of their _Centers_; nor can the _Globes_ of the _Sun_ and _Planets_ otherwise be destroy'd, but by taking from them this Power of keeping their Parts united.

The Affections or Properties of _Gravity_, and its manner of acting upon _Bodies falling_, have been in a great measure discovered, and most of them made out by _Mathematical Demonstration_ in this our _Century_, by the accurate diligence of _Galilæus_, _Torricellius_, _Hugenius_, and others, and now lately by our worthy Countryman, Mr. _Isaac Newton_, which Properties it may be very material here to enumerate, that they may serve for a Foundation to all those that shall be willing to spend their Thoughts in search of the true Cause of this _Descent of Bodies_.

The first Property is, That by this Principle of _Gravitation_, all Bodies do descend towards a Point, which either is, or else is very near to the Center of Magnitude of the Earth and Sea, about which the Sea forms it self exactly into a _Spherical Surface_, and the _Prominences_ of the Land, considering the Bulk of the whole, differ but insensibly therefrom.

_Secondly_, That this Point or Center of _Gravitation_, is fix'd within the _Earth_, or at least has been so, ever since we have any _Authentick History_: For a Consequence of its Change, though never so little, would be the over-flowing of the low Lands on that side of the _Globe_ towards which it approached, and the leaving new Islands bare on the opposite side, from which it receded; but for this Two Thousand Years it appears, that the low Islands of the _Mediterranean Sea_ (near to which the ancientest Writers liv'd) have continued much at the same height above the Water, as they now are found; and no _Inundations_ or _Recesses_ of the _Sea_ arguing any such Change, are recorded in History; excepting the _Universal Deluge_, which can no better way be accounted for, than by supposing this Center of _Gravitation_ removed for a time, towards the middle of the then inhabited Parts of the World; and a change of its Place, but the Two Thousandth Part of the _Radius_ of this _Globe_, were sufficient to bury the Tops of the highest Hills under Water.

_Thirdly_, That in all Parts of the _Surface_ of the _Earth_, or rather in all Points equidistant from its _Center_, the Force of _Gravity_ is nearly equal; so that the length of the _Pendulum_ vibrating _Seconds of Time_, is found in all Parts of the World to be very near the same. 'Tis true at St. _Helena_, in the _Latitude_ of 16 Degrees _South_, I found that the _Pendulum_ of my Clock, which vibrated _Seconds_, needed to be made shorter than it had been in _England_, by a very sensible Space (but which at that time I neglected to observe accurately) before it would keep time; and since the like Observations have been made by the _French Observers_, near the _Æquinoctial_: Yet I dare not affirm, that in mine it proceeded from any other Cause, than the great Height of my Place of Observation above the _Surface_ of the _Sea_, whereby the _Gravity_ being diminished, the length of the _Pendulum_ vibrating _Seconds_, is proportionably short'ned.

_Fourthly_, That _Gravity_ does equally affect all _Bodies_, without regard either to their _Matter_, _Bulk_, or _Figure_; so that the Impediment of the _Medium_ being removed, the most compact and most loose, the greatest and smallest _Bodies_ would descend the same _Spaces_ in equal Times; the Truth thereof will appear from the Experiment I before-cited. In these two last Particulars, is shewn the great difference between _Gravity_ and _Magnetism_, the one affecting only _Iron_, and that towards its _Poles_, the other all _Bodies_ alike in every part. As a _Corollary_, from hence it will follow, that there is no such thing as _positive Levity_, those things that appear light, being only comparatively so; and whereas several things rise and swim in _Fluids_, 'tis because, Bulk for Bulk, they are not so heavy as those _Fluids_; nor is there any Reason why _Cork_, for Instance, should be said to be light, because it swims on Water, any more than _Iron_, because it swims on _Mercury_.

_Fifthly_, That this Power increases as you descend, and decreases as you ascend from the Center, and that in the Proportion of the Squares of the _Distances_ therefrom _reciprocally_, so as at a double Distance to have but a quarter of the Force; this Property is the Principle on which Mr. _Newton_ has made out all the _Phænomena_ of the _Cœlestial Motions_, so easily and naturally, that its Truth is past Dispute. Besides that, it is highly rational, that the _attractive_ or _gravitating_ Power should exert it self more vigorously in a small Sphere, and weaker in a greater, in proportion as it is contracted or expanded; and if so, seeing that the _Surfaces_ of _Spheres_ are as the _Squares_ of their _Radii_, this Power, at several Distances, will be as the _Squares_ of those _Distances reciprocally_; and then its whole Action upon each _Spherical Surface_, be it great or small, will be always equal. And this is evidently the Rule of _Gravitation_ towards the _Centers_ of the _Sun_, _Jupiter_, _Saturn_ and the _Earth_, and thence is reasonably inferred, to be the general Principle observed by _Nature_, in all the rest of the _Cœlestial Bodies_.

These are the principal Affections of _Gravity_, from which the Rules of the _Fall_ of _Bodies_, and the _Motion_ of _Projects_ are _Mathematically_ deducible. Mr. _Isaac Newton_ has shew'd how to define the Spaces of the _Descent_ of a _Body_, let fall from any given height, down to the _Center_, supposing the _Gravitation_ to increase, as in the fifth Property; but considering the smallness of heighth, to which any _Project_ can be made ascend, and over how little an _Arch_ of the _Globe_ it can be cast by any of our _Engines_, we may well enough suppose the _Gravity_ equal throughout, and the Descents of _Projects_ in parallel Lines, which in Truth are towards the _Center_, the difference being so small as by no means to be discovered in _Practice_. The _Opposition_ of the _Air_, 'tis true, is considerable against all light Bodies moving through it, as likewise against small ones (of which more hereafter) but in great and ponderous Shot, this Impediment is found by _Experience_ but very small, and may safely be neglected.

_Propositions concerning the Descent of Heavy Bodies, and the Motion of _Projects_._

_Prop. I._ The _Velocities_ of _Falling Bodies_, are proportionate to the Times from the beginning of their _Falls_.

This follows, for that the Action of _Gravity_ being _continual_, in every Space of Time, the falling Body receives a new Impulse, equal to what it had before, in the same Space of Time, received from the same Power: For Instance, in the first Second of Time, the falling Body has acquired a _Velocity_, which in that time would carry it to a certain Distance, suppose 32 Foot, and were there no new Force, would descend at that rate with an _equable Motion_: But in the next Second of Time, the same Power of _Gravity_ continually acting thereon, superadds a new _Velocity_ equal to the former; so that at the end of two Seconds, the _Velocity_ is double to what it was at the end of the first, and after the same manner may it be proved to be triple, at the end of the third Second, and so on. Wherefore the _Velocities_ of _falling Bodies_, are proportionate to the Time of their _Falls_, _Q. E. D._

_Prop. II._ The _Spaces_ described by the Fall of a Body, are as the _Squares_ of the Times, from the beginning of the _Fall_.

_Demonstration._ Let AB (_Fig. 9. Tab. 4._) represent the _Time_ of the _Fall_ of a _Body_, BC perpendicular to AB, the _Velocity_ acquired at the end of the _Fall_, and draw the Line AC; then divide the Line AB representing the Time, into as many equal Parts as you please, as b, b, b, b, _&c._ and through these Points draw the Lines bc, bc, bc, bc, _&c._ parallel to BC, 'tis manifest that the several Lines, bc, represent the several _Velocities_ of the falling Body, in such Parts of the _Time_ as Ab is of AB, by the former Proposition. It is evident likewise, that the _Area_ ABC is the Sum of all the Lines bc being taken, according to the Method of _Indivisibles_, infinitely many; so that the _Area_ ABC represents the Sum of all the _Velocities_, between none and BC supposed infinitely many; which Sum is as the Space descended in the Time represented by AB. And by the same Reason the _Areas_ Abc, will represent the Spaces descended in the Times Ab; so then the Spaces descended in the Times AB, Ab, are as the _Areas_ of the _Triangles_ ABC, Abc, which by the 20th of the 6 of _Euclid_, are as the _Squares_ of their _Homologous Sides_ AB, Ab, that is to say, of the _Times_: Wherefore the Descents of _falling Bodies_, are as the _Squares_ of the Times of their _Fall_, _Q. E. D._

_Prop. III._ The _Velocity_ which a _falling Body_ acquires in any Space of time, is double to that, wherewith it would have moved the Space, descended by an equable Motion, in the same _time_.

_Demonstration._ Draw the Line EC parallel to AB, and AE parallel to BC in the same _Fig. 9._ and compleat the _Parallelogram_ ABCE, it is evident that the _Area_ thereof may represent the Space, a _Body_ moved equably with the _Velocity_ BC would describe in the Time AB, and the _Triangle_ ABC represents the _Space_ describ'd by the _Fall_ of a _Body_, in the same Time AB, by the second Proposition. Now the _Triangle_ ABC is half of the _Parallelogram_ ABCE, and consequently the Space described by the _Fall_, is half what would have been described by an _equable Motion_ with the _Velocity_ BC, in the same Time; wherefore the _Velocity_ BC at the end of the _Fall_, is double to that _Velocity_, which in the Time AB, would have described the _Space fallen_, represented by the _Triangle_ ABC with an _equable Motion_, _Q. E. D._

_Prop. IV._ All _Bodies_ on or near the Surface of the _Earth_, in their _Fall_, descend so, as at the end of the first Second of Time, they have described 16 Feet, 1 Inch, _London Measure_, and acquired the _Velocity_ of 32 Feet, 2 Inches, in a Second.

This is made out from the 25th Proposition of the second Part of that excellent Treatise of Mr. _Hugenius de Horologio Oscillatorio_; wherein he demonstrates the time of the least _Vibrations_ of a _Pendulum_, to be to the Time of the _Fall_ of a _Body_, from the heighth of half the length of the _Pendulum_, as the _Circumference_ of a _Circle_ to its _Diameter_; whence, as a _Corollary_, it follows, That as the _Square_ of the _Diameter_ to the _Square_ of the _Circumference_, so half the length of the _Pendulum_ vibrating _Seconds_, to the _Space_ described by the _Fall_ of a _Body_ in a _Second_ of _Time_: And the Length of the _Pendulum_ vibrating _Seconds_, being found 39, 125, or ⅛ Inches, the _Descent_ in a _Second_ will be found by the aforesaid _Analogy_ 16 Foot and 1 Inch; and, by the third Proposition, the _Velocity_ will be double thereto; and near to this it hath been found by several Experiments, which by reason of the _swiftness_ of the _Fall_, cannot so exactly determine its _Quantity_. The Demonstration of _Hugenius_ being the Conclusion of a long Train of _Consequences_, I shall for brevity sake omit; and refer you to his Book, where these things are more amply treated of.

From these Four _Propositions_, all _Questions_ concerning the _Perpendicular Fall of Bodies_, are easily _solved_, and either _Time_, _Height_, or _Velocity_ being assign'd, one may readily find the other two. From them likewise is the Doctrine of _Projects_ deducible, assuming the two following _Axioms_; _viz._ That a _Body_ set a moving, will move on continually in a right _Line_ with an _equable Motion_, unless some other Force or Impediment intervene, whereby it is accelerated, or retarded, or deflected.

_Secondly_, That a _Body_ being agitated by two _Motions_ at a time, does by their _compounded Forces_ pass through the same _Points_, as it would do, were the two _Motions divided_ and acted _successively_. As for Instance, Suppose a _Body_ moved in the Line GF, (_Fig. 1. Tab. 5._) from G to R, and there stopping, by another _Impulse_, suppose it moved in a _Space_ of _Time_ equal to the former, from R towards K, to V. I say, the _Body_ shall pass through the Point to V, though these two _several Forces_ acted both in the _same time_.

_Prop._ V. The _Motion_ of all _Projects_ is in the _Curve_ of a _Parabola_: Let the _Line_ GRF (in _Fig._ 1.) be the _Line_ in which the _Project_ is directed, and in which by the first _Axiom_ it would move equal _Spaces_ in equal _Times_, were it not deflected downwards by the Force of _Gravity_. Let GB be the _Horizontal Line_, and GC a _Perpendicular_ thereto. Then the _Line_ GRF being divided into equal Parts, answering to equal _Spaces_ of _Time_, let the _Descents_ of the _Project_ be laid down in _Lines parallel_ to GC, proportioned as the _Squares_ of the _Lines_ GS, GR, GL, GF, or as the _Squares_ of the _Times_, from S to T, from R to V, from L to X, and from F to B, and draw the _Lines_ TH, VD, XY, BC parallel to GF; I say, the Points T, V, X, B, are Points in the _Curve_ described by the _Project_, and that that _Curve_ is a _Parabola_. By the second _Axiom_, they are Points in the _Curve_; and the Parts of the _Descent_ GH, GD, GY, GC, = to ST, RV, LX, FB, being as the _Squares_ of the _Times_ (by the _Second Proposition_) that is, as the _Squares_ of the _Ordinates_, HT, DU, YX, BC, equal to GS, GR, GL, GF, the _Spaces_ measured in those Times; and there being no other _Curve_ but the _Parabola_, whose Parts of the _Diameter_ are as the _Squares_ of the _Ordinates_, it follows that the _Curve_ describ'd by a _Project_, can be no other than a _Parabola_: And saying, as RU the _Descent_ in any _time_, to GR or UD the _direct Motion_ in the same _time_, so is UD to a _third_ proportional; that _third_ will be the _Line_ call'd by all Writers of _Conicks_, the _Parameter_ of the _Parabola_ to the _Diameter_ GC, which is always the same in _Projects_ cast with the same _Velocity_: And the _Velocity_ being defined by the Number of _Feet_ moved in a _Second_ of Time, the _Parameter_ will be found by dividing the _Square_ of the _Velocity_, by 16 _Feet_, 1 _Inch_, the _Fall_ of a _Body_ in the same _Time_.

_Lemma._

The _Sine_ of the double of any _Arch_, is equal to twice the _Sine_ of that _Arch_ into its _Co-sine_, divided by _Radius_; and the _versed Sine_ of the _double_ of any _Arch_ is equal to twice the _Square_ of the _Sine_ thereof divided by _Radius_.

Let the _Arch_ BC (in _Fig. 2. Tab. 5._) be double the _Arch_ BF, and A the _Center_; draw the _Radii_ AB, AF, AC, and the _Chord_ BDC, and let fall BE perpendicular to AC, and the _Angle_ EBC, will be equal to the _Angle_ ABD, and the _Triangle_ BCE, will be like to the _Triangle_ BDA; wherefore it will be as AB to AD, so BC or twice BD, to BE; that is, as _Radius_ to _Co-sine_, so twice _Sine_ to _Sine_ of the double _Arch_. And as AB to BD, so twice BD or BC to EC, that is, as _Radius_ to _Sine_, so twice that _Sine_, to the _Versed Sine_ of the double _Arch_; which two _Analogies_ resolved into _Equations_, are the _Propositions_ contained in the _Lemma_ to be proved.

_Prop._ VI. The _Horizontal_ Distances of _Projections_ made with the same _Velocity_, at several _Elevations_ of the _Line_ of Direction, are as the _Sines_ of the doubled _Angles_ of _Elevation_.

Let GB (_Fig._ 1) the _Horizontal_ Distance be = _z_, the _Sine_ of the _Angle_ of _Elevation_, FGB, be = _s_, its _Co-sine_ = _c_, _Radius_ = _r_, and the _Parameter_ = _p_. It will be as _c_ to _s_; so _z_ to _sz_/_c_ = FB = GC, and by reason of the _Parabola_ _psz_/_c_ = to the _Square_ of CB, or GF; Now as _c_ to _r_, so is _z_ to _zr_/_c_ = GF, and its _Square_ _zzrr_/_cc_ will be therefore = to _psz_/_c_: Which _Equation_ reduced will be _psc_/_rr_ = _z_. But by the former _Lemma_ 2_sc_/_r_ is equal to the _Sine_ of the double _Angle_, whereof _s_ is the _Sine_: Wherefore 'twill be as _Radius_ to _Sine_ of double the _Angle_ FGB, so is half the _Parameter_, to the _Horizontal Range_ or _Distance_ sought; and at the several _Elevations_, the _Ranges_ are as the _Sines_ of the double _Angles_ of _Elevation_, _Q. E. D._

_Corollary._

Hence it follows, that half the _Parameter_ is the greatest _Randon_, and that that happens at the _Elevation_ of 45 Degrees, the _Sine_ of whose double is _Radius_. Likewise that the _Ranges_ equally distant above and below 45 are equal, as are the _Sines_ of all double _Arches_, to the _Sines_ of their doubled _Complements_.

_Prop._ VII. The _Altitudes_ of _Projections_ made with the same _Velocity_, at several _Elevations_, are as the _versed Sines_ of the doubled _Angles_ of _Elevation_: As _c_ is to _s_; so is _psc/rr_ = GB to _pss/rr_ = BF: and UK = RU = BF/4, the _Altitude_ of the _Projection_ = _psc/4rr_. Now by the foregoing _Lemma_ _2ss/r_ = to the _versed Sine_ of the double _Angle_, and therefore it will be as _Radius_, to _versed Sine_ of double the _Angle_ FGB, so an 8th of the _Parameter_ to the height of the _Projection_ VK; and so these heights at several _Elevations_, are as the said _versed Sines_, _Q. E. D._

_Corollary._

From hence it is plain, that the greatest _Altitude_ of the perpendicular _Projection_ is a 4th of _Parameter_, or half the greatest _Horizontal Range_; the _versed Sine_ of 180 Degrees being = _2r_.

_Prop._ VIII. The _Lines_ GF, or Times of the Flight of a _Project_ cast with the same Degree of _Velocity_ at different _Elevations_, are as the _Sines_ of the _Elevations_.

As _c_ is to _r_; so is _psc/rr_ = GB by the 6 Prop. to _ps/r_ GF; that is, as _Radius_ to _Sine_ of _Elevation_, so the _Parameter_ to the _Line_ GF; so the _Lines_ GF are as the _Sines_ of _Elevation_, and the _Times_ are proportional to those _Lines_; wherefore the _Times_ are as the _Sines_ of _Elevation_: _Ergo constat propositio_.

_Prop._ IX. Problem. A _Projection_ being made as you please, having the Distance and Altitude, or Descent, of an Object, through which the Project passes, together with the _Angle_ of _Elevation_ of the _Line_ of _Direction_; to find the _Parameter_ and _Velocity_, that is (in _Fig._ 1.) having the _Angle_ FGB, GM, and MX.

_Solution._ As _Radius_ to _Secant_ of FGB, so GM the _Distance_ given to GL; and as _Radius_ to _Tangent_ of FGB, so GM to LM. Then LM - MX in _Heights_, or + MX in _Descents_; or else MX - ML, if the _Direction_ be below the _Horizontal Line_, is the _Fall_ in the _Time_ that the direct _Impulse_ given in G would have carried the Project from G to L = LX = GY; then by Reason of the _Parabola_, as LX or GY, is to GL or YX, so is GL to the _Parameter_ sought. To find the _Velocity_ of the _Impulse_: by Prop. 2, and 4, find the Time in Seconds that a Body would fall the Space LX; and by that dividing the Line GL, the _Quote_ will be the _Velocity_, or Space moved in a Second sought, which is always a mean Proportional between the _Parameter_, and 16 Feet, 1 Inch.

_Prop_. X. Problem 2. Having the _Parameter_, Horizontal Distance, and Height or Descent of an _Object_, to find the Elevations of the Line of Direction necessary to hit the given _Object_; that is, having GM, MX, and the greatest _Randon_ equal to half the _Parameter_; to find the _Angles_ FGB.

Let the _Tangent_ of the _Angle_ sought be = _t_, the _Horizontal Distance_ GM = _b_, the Altitude of the _Object_ MX = _h_, the _Parameter_ = _p_, and _Radius_ = _r_, and it will be,

As _r_ to _t_, so _b_ to _tb/r_ = ML and _tb/r ∓ h_ {in ascents|in descents} = LX, and _ptb/r ∓ ph_ = GL _quad._ = XY _quad. ratione Parabolæ_; but _bb ∓ ttbb/rr_ = GL _quad._ 47. 1. _Euclid_. Wherefore _ptb/r ∓ ph_ = _bb ∓ ttbb/rr_ which Equation transposed, is _ttbb/rr_ = _ptb/r ∓ ph - bb_, divided by _bb_ is _tt/rr_ = _pt/br ∓ ph/bb_ - 1.

this Equation shews the Question to have 2 Answers, and the Roots thereof are

_t/r_ = _p/2h_ ∓ √((_pp ∓ 4ph_)/_4bb_) - 1;

from which I derive the following Rule.

Divide half the _Parameter_ by the Horizontal distance, and keep the Quote; _viz._ _p/2b_ then say, as _square_ of the _distance_ given to the half _Parameter_, so half _Parameter_ ∓ double {height|descent} to the _square_ of a _Secant_ = (_pp ∓ 4ph_)/(_4bb_). The _Tangent_ answering to that _Secant_, will be

√((_pp ∓ 4ph_)/4_bb_) - 1 or Square of Radius,

so then the sum and difference of the afore-found _Quote_, and this _Tangent_ will be the Roots of the _Equation_, and the _Tangents_ of the _Elevations_ sought.

Note here, that in _Descents_, if the _Tangent_ exceed the _Quote_, as it does when _ph_ is more than _bb_, the _direction_ of the lower _Elevation_ will be below the _Horizon_, and if _ph_ = _bb_, it must be directed _Horizontal_, and the _Tangent_ of the upper _Elevation_ will be _pr/b_: Note likewise, that if _4bb + 4ph_ in _Ascents_, or _4bb - 4ph_ in _Descents_, be equal to _pp_, there is but one _Elevation_ that can hit the _Object_, and its _Tangent_ is _pr/2b_. And if _4bb + 4ph_ in _Ascents_, or _4bb - 4ph_ in _descents_, do exceed _pp_, the _Object_ is without the reach of a _Project_ cast with that _Velocity_, and so the thing impossible.