Chapter X: Appendix (9)

It is about forty Years since, that the _Thermometers_ of _Robert de Fluctibus_, depending on the Dilatation and Contraction of included Air by Heat and Cold, have been disused, upon discovery that the Airs pressure is unequal; that inequality mixing it self with the Effects of the warmth of the Air in that Instrument. And instead thereof was substituted the seal'd _Thermometer_, including Spirit of Wine (first brought into _England_, out of _Italy_, by Sir _Robert Southwell_) as a proper Standard of the temper of the Air, in relation to Heat and Cold; that Ætherial Spirit being of all the known Liquors the most susceptible of Dilatation and Contraction, especially with a moderate degree of either Heat or Cold. Now this being allow'd as a Standard, and the other _Thermometer_ that includes Air, being graduated with the same Divisions, so as at the time when the Air was included, to agree with the Spirit-_Thermometer_ in all the degrees of Heat and Cold, noting at the same time the precise height of the _Mercury_ in the common Barometers: It will readily be understood, that whensoever these two _Thermometers_ shall agree, the pressure of the Air is the same it was, when the Air was included, and the Instrument graduated: That if in the Air-_Thermometer_ the Liquor stand higher than the Division marked thereon, corresponding with that on the Spirit-glass, it is an indication that there is a greater pressure of the Air at that time, than when the Instrument was graduated. And the contrary is to be concluded, when the Air-glass stands lower than the Spirit, _viz._ that then the Air is so much lighter, and the _Quick silver_, in the ordinary Barometer lower than at the said time of Graduation.

And the Spaces answering to an Inch of _Mercury_, will be more or less, according to the quantity of Air so included, and the smallness of the Glass Cane, in which the Liquor rises and falls, and may be augmented almost in any proportion, under that of the Specifick Gravity of the Liquor of the _Thermometer_ to _Mercury_. So as to have a Foot or more for an Inch of _Mercury_, which is another great convenience.

It has been observed by some, that in long keeping this Instrument, the Air included either finds a means to escape, or deposites some Vapours mixt with it, or else for some other cause becomes less Elastick, whereby, in process of time, it gives the height of the _Mercury_ somewhat greater than it ought; but this, if it should happen in some of them, hinders not the usefulness thereof, for that it may at any time very easily be corrected by Experiment, and the rising and falling thereof are the things chiefly remarkable in it, the just height being barely a Curiosity.

In these Parts of the World, long Experience has told us, that the rising of the _Mercury_ forebodes fair Weather after foul, and an Easterly or Northerly Wind; and that the falling thereof, on the contrary, signifies Southerly or Westerly Winds, with Rain, or stormy Winds, or both; which latter it is of much more consequence to provide against at Sea than at Land; and in a Storm, the _Mercury_ beginning to rise is a sure sign that it begins to abate, as has been experienced in high Latitudes, both to the Northwards and Southwards of the Æquator.

The Form of this Instrument is shown in the Cut, by Tab. 4. Fig. 1. wherein,

AB represents the Spirit-_Thermometer_, graduated from 0, or the freezing Point, through all the possible degrees of the Heat or Cold of the Air, at least in these Climates.

CD, is the Air-_Thermometer_, graduated after the same manner with the like Degrees.

EF, is a Plate applied to the side of the _Thermometer_ CD, graduated into Spaces answering to Inches and parts of an Inch of _Mercury_, in the common Barometers.

G, a Hand standing on the Plate at the height of the _Mercury_ thereon, as it was when the Instrument was graduated, as suppose here at 29½ Inches.

LM, a Wire on which the Plate EF, slips up and down, parallel to the Cane of the _Thermometer_ CD.

K, any Point at which the Spirit stands at the time of Observation; suppose at 38 on the Spirit-_Thermometer_; Slide the Plate EF till the Hand G stand at 38 on the Air-_Thermometer_, and if the Liquor therein stand at 38 likewise, then is the pressure of the Air the same as at the time of Graduation, _viz._ 29,5; but if it stand higher, as at 30, at I; then is the pressure of the Air greater; and the division on the sliding Plate against the Liquor, shews the present height of the _Mercury_ to be twenty nine Inches seven Tenths. And this may suffice as to the manner of using it.

I had one of these Barometers with me in my late Southern Voyage, and it never failed to prognostick and give early notice of all the bad Weather we had, so that I depended thereon, and made provision accordingly; and from my own Experience I conclude that a more useful Contrivance hath not for this long time been offer for the benefit of Navigation.

These Instruments are made according to the Direction of Dr. _Hook_, by Mr. _Henry Hunt_, Operator to the Royal Society, who will furnish any Gentlemen with them, and give them Directions how to use them.

_A Discourse concerning the Proportional Heat of the Sun in all
Latitudes, with the Method of collecting the same; as it was read
before the Royal Society, in one of their late Meetings. By _E.
Halley_._

There having lately arisen some Discourse about that part of the Heat of Weather, simply produced by the Action of the Sun; and I having affirmed, that if that were considered, as the only Cause of the Heat of the Weather, I saw no Reason, but that under the Pole the solstitical Day ought to be as hot as it is under the Æquinoctial, when the Sun comes vertical, or over the Zenith: For this Reason, that for all the 24 Hours of that Day under the Pole, the Sun's Beams are inclined to the Horizon, with an Angle of 23½ Degrees; and under the Æquinoctial, though he come vertical, yet he shines no more than 12 Hours, and is again 12 Hours absent; and that for 3 Hours 8 Minutes of that 12 Hours, he is not so much elevated as under the Pole; so that he is not 9 of the whole 24, higher than 'tis there, and is 15 Hours lower. Now the simple Action of the Sun is, as all other Impulses or Stroaks, more or less forceable, according to the _Sinus_ of the Angle of Incidence, or to the Perpendicular let fall on the Plain, whence the vertical Ray (being that of the greatest Heat,) being put _Radius_, the force of the Sun on the Horizontal Surface of the Earth will be to that, as the _Sinus_ of the Sun's Altitude at any other time. This being allow'd for true, it will then follow, that the time of the continuance of the Sun's shining being taken for a _Basis_, and the _Sines_ of the Sun's Altitudes erected thereon as Perpendiculars, and a Curve drawn through the Extremities of those Perpendiculars, the _Area_ comprehended shall be proportionate to the Collection of the Heat of all the Beams of the Sun in that space of time. Hence it will follow, that under the Pole the Collection of all the Heat of a tropical Day, is proportionate to a Rectangle of the _Sine_ of 23½ _gr._ into 24 Hours, or the Circumference of a Circle; that is, the _Sine_ of 23½ _gr._ being nearly 4 Tenths of _Radius_; as 3/10 into 12 Hours. Or the Polar Heat is equal to that of the Sun containing 12 Hours above the Horizon, at 53 _gr._ height, than which the Sun is not 5 Hours more elevated under the Æquinoctial.

But that this Matter may the better be understood, I have exemplified it by a Scheme, (_Tab. 4. Fig. 2_) wherein the _Area ZGHH_, is equal to the _Area_ of all the _Sines_ of the Sun's Altitude under the Æquinoctial, erected on the respective Hours from Sun-rise to the Zenith; and the _Area ♋HH♋_ is in the same proportion to the Heat of the same 6 Hours under the Pole on the Topical Day; and _⨀HHQ_, is proportional to the collected Heat of 12 Hours, or half a Day under the Pole, which space _⨀HHQ_, is visibly greater than the other _Area HZGH_, by as much as the _Area HGQ_ is greater than the _Area ZG⨀_; which, that it is so, is visible to sight, by a great excess; and so much in proportion does the Heat of the 24 Hours Sun-shine under the Pole, exceed that of the 12 Hours under the Æquinoctial: Whence, _Cæteris paribus_, it is reasonable to conclude, that were the Sun perpetually under the Tropick, the Pole would be at least as warm, as it is now under the Line it self.

But whereas the Nature of Heat is to remain in the Subject, after the Cause that heated is removed, and particularly in the Air; under the Æquinoctial, the 12 Hours absence of the Sun does very little still the Motion impressed by the part Action of his Rays, wherein Heat consists, before he arise again: But under the Pole the long absence of the Sun for 6 Months, wherein the extremity of Cold does obtain, has so chill'd the Air, that it is as it were frozen, and cannot, before the Sun has got far towards it, be any way sensible of his presence, his Beams being obstructed by thick Clouds, and perpetual Fogs and Mists, and by that Atmosphere of Cold, as the late Honourable Mr. _Boyle_ was pleased to term it, proceeding from the everlasting Ice, which in immense Quantities does chill the Neighbouring Air, and which the too soon retreat of the Sun leaves unthawed, to encrease again, during the long Winter that follows this short interval of Summer. But the differing Degrees of Heat and Cold, in differing Places, depend in great measure upon the Accidents of the Neighbourhood of high Mountains, whose height exceedingly chills the Air brought by the Winds over them; and of the Nature of the Soil, which variously retains the Heat, particularly the Sandy, which in _Africa_, _Arabia_, and generally where such Sandy Desarts are found, do make the Heat of the Summer incredible to those that have not felt it.

In the prosecution of this first Thought, I have solved the Problem generally, _viz._ to give the proportional Degree of Heat, or the Sum of all the _Sines_ of the Sun's Altitude, while he is above the Horizon in any oblique Sphere, by reducing it to the finding of the Curve Surface of a Cylindrick Hoof, or of a given part thereof.

Now this Problem is not of that difficulty as appears at first sight, for in _Tab. 4. Fig. 3._ let the Cylinder ABCD be cut obliquely with the Ellipse BKDI, and by the Center thereof H, describe the Circle IKLM; I say, the Curve Surface IKLB is equal to the Rectangle of IK and BL, or of HK and 2 BL or BC: And if there be supposed another Circle, as NQPO, cutting the said Ellipse in the Points P, Q; draw PS, QR, parallel to the Cylinders Axe, till they meet with the aforesaid Circle IKLM in the Points R, S, and draw the Lines RTS, QVP bisected in T and V. I say again, that the Curve Surface RMSQDP is equal to the Rectangle of BL or MD and RS, or of 2 BL or AD and ST or VP; and the Curve Surface QNPD is equal to RS × MD----the Arch RMS × SP, or the Arch MS × 2 SP: Or it is equal to the Surface RMSQDP, substracting the Surface RMSQNP. So likewise the Curve Surface QBPO is equal to the Sum of the Surface RMSQDP, or RS × MD, and of the Surface RLSQOP, or the Arch LS × 2 SP.

This is the most easily demonstrated from the Consideration, That the Cylindrick Surface IKLB is to the inscrib'd Spherical Surface IKLE, either in the whole, or in its Analogous Parts, as the tangent BL is to the Arch EL, and from the Demonstrations of _Archimedes de Sphæra & Cylindro, Lib. I. Prop._ XXX, and XXXVII, XXXIIX. which I shall not repeat here, but leave the Reader the pleasure of examining it himself; nor will it be amiss to consult Dr. _Barrow_'s Learned Lectures on that Book, Publish'd at _London_, _Anno 1684_, _viz._ _Probl._ IX. and the Corollaries thereof.

Now to reduce our Case of the Sum of all the _Sines_ of the Sun's Altitude in a given Declination and Latitude to the aforesaid Problem, let us consider (_Tab. 4. Fig. 4._) which is the _Analemma_ projected on the Plain of the _Meridian_, Z the Zenith, P the Pole, HH the Horizon, ææ the Æquinoctial, ♋♋, ♑♑ the two Tropicks, ♋1 the _Sine_ of the Meridian Altitude in ♋; and equal thereto, but perpendicular to the Tropick, erect ♋I, and draw the Line TI intersecting the Horizon in T, and the Hour Circle of 6, in the Point 4, and 64 shall be equal to 6R, or to the Sine of the Altitude at 6: And the like for any other Point in the Tropick, erecting a Perpendicular thereat, terminated by the Line T I: Through the Point 4 draw the Line 4, 5, 7 parallel to the Tropick, and representing a Circle equal thereto; then shall the Tropick ♋♋ in _Fig. 4._ answer to the Circle NOPQ, in _Fig. 3._ the Circle 457 shall answer the Circle IKLM, T4I shall answer to the Elliptick Segment QIBKP, 6R or 64 shall answer to SP, and 5I to BL, and the Arch ♋T, to the Arch LS, being the semidiurnal Arch in that Latitude and Declination; the _Sine_ whereof, tho' not expressible in _Fig. 4._ must be conceived as Analogous to the Line TS or UP in _Fig. 3._

The Relation between these two Figures being well understood, it will follow from what precedes, That, _the sum of the _Sines_ of the Meridian Altitudes of the Sun in the two Tropicks, (and the like for any two opposite Parallels) being multiplied by the _Sine_ of the semidiurnal Arch, will give an _Area_ Analogous to the Curve Surface RIMSQDP; and thereto adding in Summer, or substracting in Winter, the Product of the length of the semidiurnal Arch, (taken according to _Van Ceulen_'s Numbers) into the difference of the above-said _Sines_ of the Meridian Altitude: The sum in one case, and difference in another, shall be as the Aggregate of all the _Sines_ of the Sun's Altitude, during his appearance above the Horizon; and consequently of all his Heat and Action on the Plain of the Horizon in the proposed Day_. And this may also be extended to the parts of the same Day; for if the aforesaid Sum of the _Sines_ of the Meridian Altitudes, be multiplied by half the Sum of the _Sines_ of the Sun's Horary distance from Noon, when the Times are before and after Noon; or by half their difference, when both are on the same side of the Meridian; and thereto in Summer, or therefrom in Winter, be added or substracted the Product of half the Arch answerable to the proposed interval of Time, into the difference of the _Sines_ of Meridian Altitudes, the Sum in one case and Difference in the other, shall be proportional to all the Action of the Sun during that space of time.

I fore-see it will be Objected, that I take the _Radius_ of my Circle on which I erect my Perpendiculars always the same, whereas the Parallels of Declination are unequal; but to this I answer, That our said Circular Bases ought not to be Analogous to the Parallels, but to the Times of Revolution, which are equal in all of them.

It may perhaps be useful to give an Example of the Computation of this Rule, which may seem difficult to some. Let the Solstitical Heat in ♋ and ♑ be required at _London_, _Lat._ 51° 32'.

380-2'8 _Co-Lat._ 23 -30 _Decl. ⨀_ ------ 61 -58 _Sinus_ = ,882674 14 -58 _Sinus_ = ,258257 ------ _Summa_ 1,140931 _Diff._ ,624417

_Diff. Ascen._ 3300-1'1. _Arch. Semid. æstiv._ 123-11.

_Ar. Sem. hyb. 56-49. S._ ,638923 _Arch. æstiv. mensura_ 2,149955 _Arc. hyb. mensura_ ,991683

Then 1,140931 in ,836923, + 624417 in 2,149955 = 2,29734. And 1,140931 in 836929 - ,624417 in ,991638 = 33895.

So that 2,29734 will be as the Tropical Summers Day Heat, and 0,33895 as the Action of the Sun in the Day of the Winter Solstice.

After this manner I computed the following Table for every tenth Degree of Latitude, to the Æquinoctial and Tropical Sun, by which an Estimate may be made of the intermediate Degrees.

+------+--------------+--------------+--------------+ | | Sun in | Sun in | Sun in | | Lat. | ♈ ♎ | ♋ | ♑ | +------+--------------+--------------+--------------+ | 0 | 20000 | 18341 | 18341 | | 10 | 19696 | 20290 | 15834 | | 20 | 18794 | 21737 | 13166 | | 30 | 17321 | 22651 | 10124 | | 40 | 15321 | 23048 | 6944 | +------+--------------+--------------+--------------+ | 50 | 12855 | 22991 | 3798 | | 60 | 10000 | 22773 | 1075 | | 70 | 6840 | 23543 | 000 | | 80 | 3473 | 24673 | 000 | | 90 | 0000 | 25055 | 000 | +------+--------------+--------------+--------------+

Those that desire more of the Nature of this Problem, as to the Geometry thereof, would do well to compare the XIII. _Prop. Cap. V._ of the Learned Treatise, _De Calculo Centri Gravitatis_, by the Reverend Dr. _Wallis_, Published _Anno 1670_.

From this Rule there follow several Corollaries worth Note: As I. That the Æquinoctial Heat, when the Sun comes Vertical, is as twice the Square of _Radius_, which may be proposed as a Standard to compare with, in all other Cases. II. That under the Æquinoctial, the Heat is as the _Sine_ of the Sun's Declination. III. That in the Frigid Zones when the Sun sets not, the Heat is as the Circumference of a Circle into the _Sine_ of the Altitude at 6. And consequently, that in the same Latitude these Aggregates of Warmth, are as the _Sines_ of the Sun's Declinations; and in the same Declination of _Sol_, they are as the _Sines_ of the Latitude, and generally they are as the _Sines_ of the Latitude into the _Sines_ of Declination. IV. That the Æquinoctial Days Heat is every where as the Co-sine of the Latitude. V. In all places where the Sun sets, the difference between the Summer and Winter Heats, when the Declinations are contrary, is equal to a Circle into the _Sine_ of the Altitude at six in the Summer Parallel, and consequently those differences are as the _Sines_ of Latitude into, or multiplied by the _Sines_ of Declination. VI. From the Table I have added, it appears, that the Tropical Sun under the Æquinoctial, has, of all others, the least Force. Under the Pole it is greater than any other Days Heat whatsoever, being to that of the Æquinoctial as 5 to 4.

From the Table and these Corollaries may a general _Idea_ be conceived of the Sum of all the Actions of the Sun in the whole Year, and that part of the Heat that arises simply from the Presence of the Sun be brought to a Geometrical Certainty: And if the like could be performed for Cold; which is something else than the bare Absence of the Sun, as appears by many Instances, we might hope to bring what relates to this part of _Meteorology_ to a perfect Theory.

_Concerning the Distance of the Fix'd Stars. By the Honourable _Francis
Roberts_, Esq; S. R. S._

The Ancient Astronomers, who had no other way of computing the Distances of the Heavenly Bodies, but by their Parallax to the Semi-diameter of the Earth; and being never able to discover any in the fix'd Stars, did from thence rightly enough infer, that their Distance was very great, and much exceeding that of the Planets, but could go no farther otherwise than by uncertain guess.

Since the _Pythagorean_ System of the World has been reviv'd by _Copernicus_, (and now by all Mathematicians accepted for the true one) there seem'd Ground to imagine that the Diameter of the Earth's Annual Course (which, according to our best Astronomers, is at least 40000 times bigger than the Semi-diameter of the Earth) might give a sensible Parallax to the fix'd Stars, though the other could not, and thereby determine their Distance more precisely.

But though we have a Foundation to build on so vastly exceeding that of the Ancients, there are some Considerations may make us suspect that even this is not large enough for our purpose.

Monsieur _Hugens_ (who is very exact in his Astronomical Observations) tells us, he could never discover any visible Magnitude in the fix'd Stars, though he used Glasses which magnified the apparent Diameter above 100 times.

Now, since in all likelyhood the fix'd Stars are Suns, (perhaps of a different Magnitude) we may as a reasonable _Medium_ presume they are generally about the bigness of the Sun.

Let us then (for Example) suppose the Dog-Star to be so. The Distance from us to the Sun being about 100 times the Sun's Diameter (as is demonstrable from the Sun's Diameter being 32 Minutes) it is evident, that the Angle under which the Dog-Star is seen in Mr. _Hugens_'s Telescope, must be near the same with the Angle of its Parallax to the Sun's Distance, or Semi-diameter of the Earth's Annual Course; so that the Parallax to the whole Diameter, can be but double such a quantity, as even to Mr. _Hugens_'s nice Observation is altogether insensible.

The Distance therefore of the fix'd Stars seems hardly within the reach of any of our Methods to determine; but from what has been laid down, we may draw some Conclusions that will much illustrate the prodigious vastness of it.

1. That the Diameter of the Earth's Annual Orb (which contains at least 160 Millions of Miles) is but as a Point in comparison of it; at least it must be above 6000 times the Distance of the Sun. For if a Star should appear thro' the aforesaid Telescope half a Minute broad (which is a pretty sensible Magnitude) the true apparent Diameter would not exceed 18 3d Minutes, which is less than the 6000th part of the apparent Diameter of the Sun, and consequently the Sun's Distance not the 6000th part of the Distance of the Star.

2. That could we advance towards the Stars 99 Parts of the whole Distance, and have only 1/100 Part remaining, the Stars would appear little bigger to us than they do here; for they would shew no otherwise than they do through a Telescope, which magnifies an Hundred-fold.

3. That at least Nine Parts in Ten of the Space between us and the fix'd Stars, can receive no greater Light from the Sun, or any of the Stars, than what we have from the Stars in a clear Night.

4. That Light takes up more time in travelling from the Stars to us, than we in making a _West-India_ Voyage (which is ordinarily perform'd in six Weeks.) That a Sound would not arrive to us from thence in 50000 Years, nor a Cannon-bullet in a much longer time. This is easily computed, by allowing (according to Mr. _Newton_) Ten Minutes for the Journey of Light from the Sun hither, and that a Sound moves about 1300 Foot in a Second.

_The Famous Mr. _Isaac Newton_'s Theory of the _Moon_._

This _Theory_ which hath been long expected by all the true Lovers of _Astronomy_, was communicated from Mr. _Newton_ to Dr. _Gregory_, _Astronomy_ Professor at _Oxford_, and by him published in his _Astron. Elem. Philos._ and _Geomet._ p. 336. From whence, as it was lately translated into _English_, I thought fit to insert it here.

By this Theory, what by all Astronomers was thought most difficult and almost impossible to be done, the Excellent Mr. _Newton_ hath now effected; _viz._ to determine the Moon's Place even in her Quadratures, and all other Parts of her Orbit, besides the Syzygys, so accurately by Calculation, that the Difference between that and her true Place in the Heavens, shall scarce be above two minutes in her Syzygys, or above three in her Quadratures, and is usually so small, that it may well enough be reckon'd only as a Defect in the Observation. And this Mr. _Newton_ experienced, by comparing it with very many Places of the Moon, observ'd by Mr. _Flamsteed_, and communicated to him.

The Royal Observatory at _Greenwich_, is to the West of the Meridian of _Paris_, 2 degrees, 19 minutes. Of _Uraniburgh_, 12 degrees, 51 minutes, 30 seconds. And of _Gedanum_, 18 degrees, 48 minutes.

The mean Motions of the Sun and Moon, accounted from the Vernal Æquinox at the Meridian of _Greenwich_, I make to be as followeth.

The last Day of _December 1680_, at Noon (_Old Stile_) the mean Motion of the Sun was 9 Signs, 20 degrees, 34 minutes, 46 seconds. Of the Sun's Apogæum, was 3 S. 7 deg. 23 min. 30 seconds.

That the mean Motion of the Moon at that time, was 6 S. 1 degree, 45 minutes, 45 seconds. And of her Apogee, 8 S. 4 degrees, 28 minutes, 5 seconds. Of the ascending Node of the Moon's Orbit, 5 S. 24 deg. 14 min. 35 seconds, _&c._

And on the last Day of _December, 1700_, at Noon, the mean Motion of the Sun was 9 S. 20 degrees, 43 minutes, 50 seconds. Of the Sun's Apogee, 3 S. 7 degrees, 44 minutes, 30 seconds. The mean Motion of the Moon was 10 S. 15 degrees, 19 minutes, 50 seconds. Of the Moon's Apogee, 11 S. 8 degrees, 18 minutes, 20 seconds. And of her ascending Node, 4 S. 27 degrees, 24 minutes, 20 seconds. For in 20 _Julian_ Years, or 7305 Days, the Sun's Motion is 20 Revol. 0 S. 0 degrees, 9 minutes, 4 seconds. And the Motion of the Sun's Apogee, 21 minutes, 0 seconds.

The Motion of the Moon in the same time, is 267 Revol. 4 S. 13 degrees, 34 minutes, 5 seconds. And the Motion of the Lunar Apogee, is 2 Revol. 3 S. 3 degrees, 50 minutes, 15 seconds. And the Motion of her Node, 1 Revol. 0 S, 26 degrees, 50 minutes, 15 seconds.

All which Motions are accounted from the Vernal Æquinox: Wherefore if from them there be subtracted the Recession or Motion of the Æquinoctial Point, in _Antecedentia_, during that space, which is 16 minutes, 40 seconds, there will remain the Motions in reference to the fix'd Stars in 20 _Julian_ Years; _viz._ the Sun's 19 Revol. 11 S. 29 degrees, 52 minutes, 24 seconds. Of his Apogee, 4 minutes, 20 seconds. And the Moon's 267 Revol. 4 S. 13 degrees, 17 minutes, 25 seconds. Of her Apogee, 2 Revol. 3 S. 3 degrees, 33 minutes, 35 seconds. And of the Node of the Moon, 1 Revol. 0 S. 27 degrees, 6 minutes, 55 seconds.

According to this Computation, the _Tropical Year_ is 365 Days, 5 Hours, 48 Minutes, 57 Seconds. And the _Sydereal Year_ is 365 Days, 6 Hours, 9 Minutes, 14 Seconds.

These mean Motions of the Luminaries are affected with various Inequalities: Of which,

1. There are the Annual Equations of the aforesaid mean Motions of the Sun and Moon, and of the Apogee and Node of the Moon.

The Annual Equation of the mean Motion of the Sun, depends on the Eccentricity of the Earth's Orbit round the Sun, which is 16-11/12 of such Parts, as that the Earth's mean Distance from the Sun shall be 1000: Whence 'tis call'd the _Equation of the Centre_; and is, when greatest, 1 degree, 56 minutes, 20 seconds.

The greatest Annual Equation of the Moon's mean Motion, is 11 degrees, 49 seconds; of her Apogee, 20 minutes, and of her Node, 9 minutes, 30 seconds.

And these four Annual Equations are always mutually proportional one to another: Wherefore when any of them is at the greatest, the other three will also be greatest; and when any one lessens, the other three will also be diminished in the same Ratio.

The Annual Equation of the Sun's Centre being given, the three other corresponding Annual Equations will be also given; and therefore a Table of that will serve for all. For if the Annual Equation of the Sun's Centre be taken from thence, for any Time, and be call'd P, and let 1/10P = Q, Q + 1/60Q = R, 1/6P = D, D + 1/30D = E, and D - 1/50D = 2F; then shall the Annual Equation of the Moon's mean Motion for that time be R, that of the Apogee of the Moon will be E, and that of the Node F.

Only observe here, That if the Equation of the Sun's Centre be required to be added; then the Equation of the Moon's mean Motion must be subtracted, that of her Apogee must be added, and that of the Node subducted, And on the contrary, if the Equation of the Sun's Centre were to be subducted, the Moon's Equation must be added, the Equation of her Apogee subducted, and that of her Node added.

There is also an _Equation of the Moon's mean Motion_, depending on the situation of her Apogee, in respect of the Sun; which is greatest when the Moon's Apogee is in an Octant with the Sun, and is nothing at all when it is in the Quadratures or Syzygys. This Equation, when greatest, and the Sun in _Perigæo_, is 3 Minutes, 56 Seconds. But if the Sun be in _Apogæo_, it will never be above 3 Minutes, 34 Seconds. At other Distances of the Sun from the Earth, this Equation, when greatest, is reciprocally as the Cube of such Distance. But when the Moon's Apogee is any where but in the _Octants_, this Equation grows less, and is mostly at the same distance between the Earth and Sun, as the Sine of the double Distance of the Moon's Apogee, from the next Quadrature or Syzygy, to the Radius.

This is to be added to the Moon's Motion, while her Apogee passes from a Quadrature with the Sun to a Syzygy; but this is to be subtracted from it, while the Apogee moves from the Syzygy to the Quadrature.

There is moreover another _Equation of the Moon's Motion_, which depends on the Aspect of the Nodes of the Moon's Orbit with the Sun: And this is greatest, when her Nodes are in _Octants_ to the Sun, and vanishes quite, when they come to their Quadratures or Syzygys. This Equation is proportional to the Sine of the double Distance of the Node from the next Syzygy, or Quadrature; and at greatest, is but 47 seconds. This must be added to the Moon's mean Motion, while the Nodes are passing from their Syzygys with the Sun, to their Quadratures with him; but subtracted while they pass from the Quadratures to the Syzygys.

From the Sun's true Place, take the equated mean Motion of the Lunar Apogee, as was above shew'd, the Remainder will be the Annual Argument of the said Apogee. From whence the _Eccentricity of the Moon_, and the _second Equation_ of her Apogee may be computed after the manner of the following (_which takes place also in the Computation of any other intermediate Equations_).

Tab. 3. Fig. 6. Let T represent the Earth, TS, a Right Line joining the Earth and Sun, TACB, a Right Line drawn from the Earth to the middle or mean Place of the Moon's Apogee, equated, as above: Let the Angle STA be the Annual Argument of the aforesaid Apogee, TA the least Eccentricity of the Moon's Orbit, TB the greatest. Bissect AB in G; and on the Centre C, with the Distance AC describe a Circle AFB, and make the Angle BCF = to the double of the Annual Argument. Draw the Right Line TF, that shall be the Eccentricity of the Moon's Orbit; and the Angle BTF, is the second Equation of the Moon's Apogee required.

In order to whose Determination, let the mean Distance of the Earth from the Moon, or the Semi-diameter of the Moon's Orbit, be 100000; then shall its greatest Eccentricity TB be 66782 such Parts; and the least TA, 43319. So that the greatest Equation of the Orbit, _viz._ when the Apogee is in the Syzygys, will be 7 degrees, 39 minutes, 30 seconds, or perhaps 7 degrees, 40 minutes, (for I suspect there will be some Alteration, according to the Position of the Apogee in _Cancer_ and _Capricorn_.) But when it is Quadrate to the Sun, the greatest Equation aforesaid will be 4 degrees, 57 minutes, 56 seconds; and the greatest Equation of the Apogee, 12 degrees, 15 minutes, 4 seconds.

Having from these Principles made a Table of the Equation of the Moon's Apogee, and of the Eccentricities of her Orbit to each degree of the Annual Argument, from whence the Eccentricity TF, and the Angle BTF (_viz._ the second and the principal Equation of the Apogee) may easily be had for any Time required; let the Equation thus found be added to the first Equated Place of the Moon's Apogee, if the Annual Argument be less than 90 degrees, or greater than 180 degrees, and less than 270; otherwise it must be subducted from it; and the Sum or Difference shall be the Place of the Lunar Apogee secondarily equated; which being taken from the Moon's Place equated a third time, shall leave the mean Anomaly of the Moon corresponding to any given Time. Moreover, from this mean Anomaly of the Moon, and the before-found Eccentricity of her Orbit, may be found (by means of a Table of Equations of the Moon's Centre made to every degree of the mean Anomaly, and some Eccentricities, _viz._ 45000, 50000, 55000, 60000, and 65000) the _Prostaphæresis_, or Equation of the Moon's Centre, as in the common way: And this being taken from the former Semi-circle of the middle Anomaly, and added in the latter to the Moon's Place thus thrice equated, will produce the Place of the Moon a fourth time equated.

The greatest Variation of the Moon (_viz._ that which happens when the Moon is in an Octant with the Sun) is nearly, reciprocally as the Cube of the Distance of the Sun from the Earth. Let that be taken 37 minutes, 25 seconds, when the Sun is _in Perigæo_, and 33 minutes, 40 seconds, when he is _in Apogæo_: And let the Differences of this Variation in the Octants be made reciprocally, as the Cubes of the Distances of the Sun from the Earth; and so let a Table be made of the aforesaid Variation of the Moon in her Octants (or its Logarithms) to every Tenth, Sixth, or Fifth Degree of the mean Anomaly: And for the Variation out of the Octants, make, as Radius to the Sine of the double Distance of the Moon from the next Syzygy, or Quadrature :: so let the afore-found Variation in the Octant be to the Variation congruous to any other Aspect; and this added to the Moon's Place before found in the first and third Quadrant (accounting from the Sun) or subducted from it in the second and fourth, will give the Moon's Place equated a fifth time.

Again, as Radius to the Sine of the Summ of the Distances of the Moon from the Sun, and of her Apogee from the Sun's Apogee (or the Sine of the Excess of that Summ above 360 degrees,) :: so is 2 minutes, 10 seconds, to a sixth Equation of the Moon's Place, which must be subtracted, if the aforesaid Summ or Excess be less than a Semi-circle; but added, if it be greater. Let it be made also, as Radius to the Sine of the Moon's distance from the Sun :: so 2 degrees, 20 secants, to a seventh Equation; which when the Moon's Light is increasing, add; but when decreasing, subtract; and the Moon's Place will be equated a seventh time, and this is her Place _in her proper Orbit_.

Note here, the Equation thus produced by the mean Quantity 2 degrees, 20 seconds, is not always of the same magnitude; but is increased and diminished, according to the Position of the Lunar Apogee. For if the Moon's Apogee be in Conjunction with the Sun's, the aforesaid Equation is about 54 seconds greater: But when the Apogees are in Opposition, 'tis about as much less; and it librates between its greatest Quantity 3 minutes, 14 seconds, and its least, 1 minute, 26 seconds. And this is, when the Lunar Apogee is in Conjunction, or Opposition with the Sun's: But in the Quadratures, the aforesaid Equation is to be lessen'd about 50 seconds, or 1 minute, when the Apogees of the Sun and Moon are in Conjunction; but if they are in Opposition, for want of a sufficient number of Observations, I cannot determine, whether it is to be lessen'd or increas'd. And even as to the Argument or Decrement of the Equation, 2 minutes, 20 seconds, above mentioned, I dare determine nothing certain, for the same Reason, _viz._ the want of Observations accurately made.

If the sixth and seventh Equations are augmented or diminished in a reciprocal _Ratio_ of the distance of the Moon from the Earth; _i. e._ in a direct _Ratio_ of the Moon's Horizontal Parallax, they will become more accurate: And this may be readily done, if Tables are first made to each minute of the said Parallax, and to every sixth or fifth degree of the Argument of the sixth Equation for the Sixth, as of the distance of the Moon from the Sun, for the Seventh Equation.

From the Sun's Place, take the mean motion of the Moon's ascending Node, equated as above; the Remainder shall be the Annual Argument of the Node, whence its second Equation may be computed after the following manner in the preceding Figure.

Let T, as before, represent the Earth; TS a Right Line, conjoining the Earth and Sun: Let also the Line TACB, be drawn to the Place of the ascending Node of the Moon, as above equated; and let STA be the Annual Argument of the Node. Take TA from a Scale, and let it be to AB :: as 56 to 3, or as 11⅔ to 1. Then bissect BA in C, and on C as a Centre, with the Distance CA, describe a Circle, as AFB, and make the Angle BCF, equal to double the Annual Argument of the Node before-found: So shall the Angle BTF, be the second Equation of the ascending Node; which must be added, when the Node is passing from the Quadrature to a Syzygy with the Sun; and subducted, when the Node moves from a Syzygy towards a Quadrature. By which means, the true Place of the Node of the Lunar Orbit will be gained: Whence from Tables made after the common way, the _Moon's Latitude, and the Reduction of her Orbit to the Ecliptick_, may be computed, supposing the Inclination of the Moon's Orbit to the Ecliptick, to be 4 degrees, 59 minutes, 35 seconds, when the Nodes are in Quadrature with the Sun; and 5 degrees, 17 minutes, 20 seconds, when they are in the Syzygys.

And from the Longitude and Latitude thus found, and the given Obliquity of the Ecliptick, 23 degrees, 29 minutes, the Right Ascension and Declination of the Moon will be found.

The Horizontal Parallax of the Moon, when she is in the Syzygys, at a mean distance from the Earth, I make to be 57 minutes, 30 seconds; and her Horary Motion, 33 minutes, 32 seconds, 32 thirds; and her apparent Diameter 31 minutes, 30 seconds. But in her Quadratures at a mean Distance from the Earth, I make the Horizontal Parallax of the Moon to be 59 minutes, 40 seconds, her Horary Motion 32 minutes, 12 seconds, 2 thirds, and her apparent Diameter, 31 minutes, 3 seconds. The Moon in an Octant to the Sun, and at a mean distance, hath her Centre distant from the Centre of the Earth about 60-2/9 of the Earth's Semi-diameters.

The Sun's Horizontal Parallax I make to be 10 seconds, and its apparent Diameter at a mean distance from the Earth, I make 32 minutes, 15 seconds.

The Atmosphere of the Earth, by dispersing and refracting the Sun's Light, casts a Shadow, as if it were an Opake Body, at least to the height of 40 or 50 Geographical Miles (by a Geographical Mile, I mean the sixtieth part of a Degree of a great Circle, on the Earth's Surface.) This Shadow falling upon the Moon in a Lunar Eclipse, makes the Earth's Shadow be the larger or broader. And to each Mile of the Earth's Atmosphere, is correspondent a Second in the Moon's Disk, so that the Semi-diameter of the Earth's shadow projected upon the Disk of the Moon, is to be increased about 50 seconds: Or, which is all one, in a Lunar Eclipse, the Horizontal Parallax of the Moon is to be increased in the Ratio of about 70 to 69.

Thus far the Theory of this Incomparable Mathematician. And if we had many Places of the Moon accurately observ'd, especially about her Quadratures, and these well compar'd with her Places, at the same time calculated according to this Theory; it would then appear, whether there yet remain any other sensible Equations; which when accounted for, might serve to improve and enlarge this Theory.

_An Estimate of the Degrees of the _Mortality_ of Mankind, drawn from
curious _Tables_ of the _Births_ and _Funerals_ at the City of
_Breslaw_; with an Attempt to ascertain the Price of _Annuities_ upon
_Lives_. By Mr. _E. Halley_, R. S. S._

The Contemplation of the _Mortality_ of _Mankind_, has besides the _Moral_, its _Physical_ and _Political_ Uses, both which have been some Years since most judiciously consider'd by the Curious Sir _William Petty_, in his _Natural_ and _Political_ Observations on the Bills of _Mortality_ of _London_, own'd by Captain _John Graunt_: And since in a like Treatise on the Bills of _Mortality_ of _Dublin_. But the Deduction from those Bills of _Mortality_ seemed even to their Authors to be defective: First, In that the _Number_ of the People was wanting. Secondly, That the _Ages_ of the People dying was not to be had. And Lastly, That both _London_ and _Dublin_, by reason of the great and casual Accession of _Strangers_ who die therein, (as appeared in both, by the great Excess of the _Funerals_ above the _Births_) rendred them incapable of being Standards for this purpose; which requires, if it were possible, that the People we treat of, should not at all be changed, but die where they were born, without any adventitious Increase from Abroad, or Decay by Migration elsewhere.

This _Defect_ seems in a great measure to be satisfied by the late curious Tables of the Bills of _Mortality_ at the City of _Breslaw_, lately communicated to this Honourable Society by Mr. _Justell_, wherein both the Ages and Sexes of all that die, are Monthly delivered, and compared with the number of the _Births_, for Five Years last past, _viz._ 1687, 88, 89, 90, 91, seeming to be done with all the Exactness and Sincerity possible.

This City of _Breslaw_ is the Capital City of the Province of _Silesia_; or, as the _Germans_ call it, _Schlesia_, and is situated on the Western Bank of the River _Oder_, anciently call'd _Viadrus_, near the Confines of _Germany_ and _Poland_, and very nigh the Latitude of _London_. It is very far from the Sea, and as much a _Mediterranean_ Place as can be desired, whence the Confluence of Strangers is but small, and the Manufacture of Linnen employs chiefly the poor People of the Place, as well as of the Country round about; whence comes that sort of Linnen we usually call your _Sclesiæ Linnen_; which is the chief, if not the only Merchandize of the Place. For these Reasons, the People of this City seem most proper for a _Standard_; and the rather, for that the _Births_ do a small matter exceed the Funerals. The only thing wanting, is the Number of the whole People, which in some measure I have endeavour'd to supply, by the comparison of the _Mortality_ of the People of all Ages, which I shall from the said Bills trace out with all the Accuracy possible.

It appears that in the Five Years mentioned, _viz._ from 87 to 91 inclusive, there were born 6193 Persons, and buried 5869; that is, born _per Annum_ 1238, and buried 1174; whence an _Increase_ of the People may be argued of 64 _per Annum_, or of about a 20th part, which may perhaps be balanc'd by the Levies for the _Emperor_'s Service in his Wars. But this being contingent, and the Births certain, I will suppose the People of _Breslaw_ to be increased by 1238 _Births_ annually. Of these it appears by the same Tables, that 348 do die _yearly_ in the _first Year_ of their _Age_, and that but 890 do arrive at a full _Year's Age_; and likewise, that 198 do die in the _Five Years_ between 1 and 6 compleat, taken at a _Medium_; so that but 692 of the Persons _born_ do survive _Six_ whole _Years_. From this _Age_ the Infants being arrived at some degree of Firmness, grow less and less _Mortal_; and it appears, that of the whole People of _Breslaw_ there die _yearly_, as in the following Table, wherein the upper Line shews the _Age_, and the next under it, the _Number_ of Persons of that Age _dying yearly_.

7 8 9 14 18 21 27 28 35
11 11 6 5½ 2 3½ 5 6 4½ 6½ 9 8 7 7

36 42 45 39 54 55 56 63
8 9½ 8 9 7 7 10 11 9 9 10 12

70 71 72 77 81 84 90 91
9½ 14 9 11 9½ 6 7 3 4 2 1 1 1

98 99 100
0 ⅕ ⅗

And where no Figure is placed over, it is to be understood of those that die between the Ages of the precedent and consequent _Column_.

From this Table it is evident, that from the Age of 9 to about 25, there does not die above 6 _per Annum_ of each Age, which is much about 1 _per Cent._ of those that are of those _Ages_: And whereas in the 14, 15, 16, 17 _Years_, there appear to die much fewer, as 2 and 3½; yet that seems rather to be attributed to Chance, as are the other Irregularities in the Series of Ages, which would rectifie themselves, were the number of Years much more considerable, as 20 instead of 5. And by our own Experience in _Christ-Church Hospital_, I am inform'd there die of the _Young Lads_, much about 1 _per Cent. per Annum_, they being of the aforesaid _Ages_. From 25 to 50, there seem to die from 7 to 8 and 9 _per Annum_ of each Age; and after that to 70, they growing more _crasie_, though the number be much diminished, yet the _Mortality increases_, and there are found to die 10 or 11 of each Age _per Annum_: From thence the number of the Living being grown very small, they gradually decline till there be none left to _die_; as may be seen at one View in the Table.

From these Considerations I have form'd the _adjoined Table_, whose Uses are manifold, and give a more just _Idea_ of the _State_ and _Condition_ of _Mankind_, than any thing yet extant that I know of. It exhibits the _Number_ of _People_ in the City of _Breslaw_ of all Ages, from the _Birth_ to extreme _Old Age_, and thereby shews the Chances of _Mortality_ at all _Ages_, and likewise how to make a certain Estimate of the Value of _Annuities_ for _Lives_, which hitherto has been only done by an imaginary _Valuation_: Also the _Chances_ that there are that a _Person_ of any _Age_ proposed does live to any other _Age_ given; with many more, as I shall hereafter shew. This _Table_ does shew the _Number_ of Persons that are living in the _Age_ current annexed thereto, as follows:

+------+---------+------+---------+------+---------+ | Age. | Persons.| Age. | Persons.| Age. | Persons.| | Curt.| | Curt.| | Curt.| | +------+---------+------+---------+------+---------+ | 1 | 1000 | 8 | 680 | 15 | 628 | | 2 | 855 | 9 | 670 | 16 | 622 | | 3 | 798 | 10 | 661 | 17 | 616 | | 4 | 760 | 11 | 653 | 18 | 610 | | 5 | 732 | 12 | 646 | 19 | 604 | | 6 | 710 | 13 | 640 | 20 | 598 | | 7 | 692 | 14 | 634 | 21 | 592 | +------+---------+------+---------+------+---------+ | 22 | 586 | 29 | 539 | 36 | 481 | | 23 | 579 | 30 | 531 | 37 | 472 | | 24 | 573 | 31 | 523 | 38 | 463 | | 25 | 567 | 32 | 515 | 39 | 454 | | 26 | 560 | 33 | 507 | 40 | 445 | | 27 | 553 | 34 | 499 | 41 | 436 | | 28 | 546 | 35 | 490 | 42 | 427 | +------+---------+------+---------+------+---------+ | 43 | 417 | 50 | 346 | 57 | 272 | | 44 | 407 | 51 | 335 | 58 | 262 | | 45 | 397 | 52 | 324 | 59 | 252 | | 46 | 387 | 53 | 313 | 60 | 242 | | 47 | 377 | 54 | 302 | 61 | 232 | | 48 | 367 | 55 | 292 | 62 | 222 | | 49 | 357 | 56 | 282 | 63 | 212 | +------+---------+------+---------+------+---------+ | 64 | 202 | 71 | 131 | 78 | 58 | | 65 | 192 | 72 | 120 | 79 | 49 | | 66 | 182 | 73 | 109 | 80 | 41 | | 67 | 172 | 74 | 98 | 81 | 34 | | 68 | 162 | 75 | 88 | 82 | 28 | | 69 | 152 | 76 | 78 | 83 | 23 | | 79 | 142 | 77 | 68 | 84 | 20 | +------+---------+------+---------+------+---------+

Age. Persons.

7 5547
14 4584
21 4270
28 3964
35 3604
42 3708
49 2709
56 2194
63 1694
70 1204
77 692
84 253
100 107
-----------
34000
-----------
Sum Total.

Thus it appears, that the whole People of _Breslaw_ does consist of 34000 _Souls_, being the Sum _Total_ of the Persons of all Ages in the _Table_: The first use hereof is to shew the Proportion of _Men_ able to bear _Arms_ in any _Multitude_, which are those between 18 and 56, rather than 16 and 60; the one being generally too weak to bear the _Fatigues_ of _War_, and the Weight of _Arms_; and the other too crasie and infirm from _Age_, notwithstanding particular Instances to the contrary. Under 18 from the _Table_, are found in this City 11997 Persons, 3950 above 56, which together make 15947, so that the Residue to 34000 being 18053, are Persons between those _Ages_. At least one half thereof are Males, or 9027: So that the whole Force this City can raise of _Fencible Men_, as the _Scotch_ call them, is about 9000, or 9/34, or somewhat more than a quarter of the _Number_ of _Souls_; which may parhaps pass for a Rule for all other places.

The _Second Use_ of this _Table_, is, to shew the differing degrees of _Mortality_, or rather _Vitality_, in all _Ages_; for if the Number of Persons of any _Age_ remaining after one Year, be divided by the difference between that and the number of the _Age_ proposed, it shews the _Odds_ that there is, that a Person of that _Age_ does not die in a _Year_. As for Instance, a Person of 25 _Years_ of _Age_ has the Odds of 560 to 7, or 80 to 1, that he does not _die_ in a _Year_: Because that of 567, living of 25 _Years_ of _Age_, there do die no more than 7 in a _Year_, leaving 560 of 26 Years old.