us sometimes swift and at other times more slow nor are the Dayes themselves accounted from Noon to Noon of equal lengths some Dayes containing more Time than some and others less there being a natural necessity that the unequal Motions of the Sun should produce such inequalities in the lengths of those natural Days that are bounded by it For a natural Day being composed of that space of Time in which any one Place or Point of the Earth is moved in its diurnal Motion East-ward from the Meridian Sun of any one Day to that of the next it will follow that these Dayes can never be equal unless the Sun in that space of Time be so mov'd in her annual Orb as to cut out equal Divisions on the Aequinoctial upon the Meridian of every Day which Divisions so intersected are by the Learned termed the Right Ascensions For whenever the Right Ascension either of Sun or Star is mentioned we are to understand by it those Degrees of the Aequinoctial that are intersected by that Meridian on which either Sun or Star have then their place But that the Sun between each Meridian does not move such just and certain Spaces in her own Orb as thus to intersect equal Divisions on the Aequi-nox upon every Meridian needs no other evidence than what either Calculation it self affords or Globes by an occular Inspection demonstrate to us By Calculation if an exact Table of Right Ascensions be composed for the Meridian or Noon-time of each particular Day there will be found almost a continual difference in the length of those Intersections that are made by the Sun on the Aequi-nox upon every Meridian so that there will by this means be found nothing but an almost continual unequalness in the Right Ascensions Which will be the more apparent if you make an Estimation of the Right Ascensions of about 10 Dayes together and compare that with those of the same Number The like will appear plainly if tryed on the Globe for if you mark out on the Ecliptick any 10 or more Day 's Motion of the Sun according to his true place found out in an exact Ephimeris and passing these 10 or more Days motion under the Meridian noting what Degrees on the Aequmoctial are then traced out which compared with the Degrees traced out by making the same number of Day 's Motion in some other part of the Ecliptick to pass the Meridian and the difference of Right Ascensions included between those two equal number of Dayes will plainly appear All which Irregularities or difference in Right Ascension proceed from two principal Causes 1. From the different Positions And 2. From the different Centres of those Orbs in which and according to which the Sun and Earth do move from whence arises a natural necessity that between two such regular and equal Motions whose Position is thus oblique those appearing differences should still arise for though both Sun and Earth the one in his annual the other in it's diurnal Revolutions be rationally supposed to be regular and equal in their own Motions yet in regard of the different Positions of their Spheres the Right Ascensions that are made by them cannot be equal it being impossible that the Sun when near Aries and Libra where he moves cross the Equinoctial should then in any particular number of days make so great an alteration in Right Ascension as he must do near the two Tropicks where both Equinoctial and Ecliptick run paralel one to the other and accordingly by the best Tables of Right Ascension 't is found that the Right Ascensions of 10 daies motion of the Sun near the Tropick of Capricorn shall arise to above 11 degrees 30 minutes whereas that of the same number of daies near the Equinoctial Point of Aries shall scarcely amount to 9 degrees Moreover from the eccentricity of these Orbs another irregularity does happen in the Right Ascensions for the Centre of the Earth upon which it turns round in it's daily Revolutions being not the same with the Centre of the Suns Orb it follows that the apparent Equinoxes pointed out by an imaginary line drawn through the centre of the Earth and intersecting the Ecliptick shall divide that Circle into two unequal parts from whence it arises that the Sun must spend more daies in passing through one half of the Ecliptick than he does in passing through the other and accordingly by experience he is found to move through that part between Libra and Aries in 179 daies but in passing between Aries and Libra he takes up 186 which is 7 dayes more so that in that part of the Year between September and March he seems to us to be swift in motion but in the other part between March and September his apparent course is more slow which seeming swiftness and slowness of the Suns motion is the cause that the Right Ascensions near both the Tropicks are not alike but differ much as do also those that are nigh the Equinoxes for the Right Ascension of 10 daies motion near the Winter Tropick is more by 60 Minutes than that of the same number of daies near that of the Summer one so also the Right Ascension of 10 daies time near Libra amounts to above 30 Minutes more than those of the same number of daies near Aries does Having thus plainly demonstrated that natural daies must needs be unequal and laid down the Causes from whence those unequalities do still arise I suppose it may now be concluded to be extreamly unreasonable for those that are so Nice and Curious as some are to expect an exact correspondence between the times given by the motions of a Clock and those divisions of it that are made by these unsteady motions of the Sun on the Lines of a Dial for if from the Reasons before laid down there be in nature a necessity for those differences of Right Ascension before asserted and that the daies which they bound must differ also in length correspondent to what those differences in Right Ascension do amount to how then is it possible that those exact and regular motions of a Pendulum to what pitch soever it be set should agree with these motions of the Sun and truely divide those daies that are not so regular as it self is For Suppose a Clock should be adjusted to the hour at a time when natural daies are shortest as about the middle of March this Clock with the same pitch of motion shall in June or December finish it's diurnal Revolutions sooner than the day shall do by reason the natural daies are now longer than those of March to which the Clock had been formerly adjusted and by consequence it shall now gain upon every day just so much time as these daies in December are longer than those of March So on the contrary if a Clock be adjusted to go true with the Sun in the Month of December at which time the natural day is alwayes longest this Clock when natural daies are
shorter as in March or September shall not finish it 's daily Revolutions so soon as the day it self shall be accomplished and by consequence go each day so much too slow as those daies of March or September are more short than them to which the Clock before had been exactly adjusted Since therefore there is no tolerable exactness in thus adjusting Clocks to the Sun it self because being thus adjusted at times when daies are either shortest or longest their gaining or losing will be the more extream in the contrary parts of the year for Example Clocks adjusted to the Sun in March shall upon most daies in December gain almost 50 Seconds which in the Months time shall amount to near half an hour and on the contrary if adjusted to go true with the Sun in December it shall in March lose the same time and so for any other according as daies do differ in length That Clocks therefore may be reduced to a more exact pitch of motion that their gain or loss may never be so extream it will be necessary to adjust them so as that their motion may be agreeable to that of a middle day or such a one as is a mean between natural daies that are most long and those others that are most short to which pitch if a Clock be once adjusted it 's gain or loss shall then be the less sensible for gain and lose it will still amounting in December but to about 15 Minutes in the whole Month and in March to but about 9 which is vastly more exact than when it shall happen to be adjusted to the longest or shortest of Natural daies or to any other that is not equal to a mean or middle day of which there are but few which in the Table are exprest by the Character ☉ Sol. But to this exact pitch of motion that may thus correspond to a mean day the greatest exactness that a Pendulum is capable of being brought to there is no way certainly to adjust a Clock without the help and assistance of a Table of Equations that give the daily differences between a mean day and those which are either longer or shorter than the mean day is which Equations having formerly been computed by the Worthy and Ingenious Mr. Christian Hugens de Zulechim who is reported to be the first that ever applyed the Pendulum to regulate the motion of a Clock and not long since Printed in number 49. Philos Trans I have made bold in regard of its exactness to transcribe in it's more natural form of an Equation by only expressing the Equations themselves without adding them together and substracting as Mr. Hugens has done for a particular use to shew the nature of a Pendulums going when set right the first of February and let go the whole year round without setting afterwards Now for their sakes that desire to know the manner of Composing such a Table themselves that thereby they may the better understand the nature of it they may Note that the Equations are to be found out and a Table composed in the manner following First find out a mean Right Ascension by dividing the 360 degrees of the Equinox into 365 parts and a quarter equal to the daies of a year and the product shall be the mean Right Ascension desired which will be found to be about three Minutes 56 Seconds according to Sir Jonas Moors account of it in his Mathematical Compendium then by the help of an exact Ephemerides here lyes the difficulty let the natural Right Ascensions of the Sun be computed by Calculation for the Meridian Position of the Sun for every day to Minutes and Seconds which having done compare the daily differences of these Natural Right Ascensions with the mean one by still substracting the lesser from the greater and what remains shall be the Equations desired still noting down either the excess or defect that is whether the natural be more than the mean or less as for example Suppose the Right Ascension between the Meridians of the 1 st and 2 d. of January be found to amount to 4 Minutes 20 Seconds this compared with the mean Right Ascensions 3 Minutes 56 Seconds and by substracting the lesser from the greater the remainder will be found to be 24 Seconds and so much the Natural Right Ascension does then exceed the mean one this 24 Seconds is the Equation for that day it being from noon to noon 24 Seconds longer than a mean day is and shews you that a Clock when well adjusted to a mean day shall then gain 24 Seconds because it finishes it's Diurnal Revolutions sooner by 24 Seconds than the day it self does on the contrary when the Right Ascensions of Natural daies are less than the mean ones as they are about the middle of March by almost 20 Seconds this 20 Seconds being the Equations belonging to such a day shall shew you that upon such a like day a well adjusted Clock shall then lose 20 Seconds for the mean day to which it is adjusted being longer than the natural one by 20 Seconds the Natural Day shall be finished sooner by 20 Seconds than the Clock at that time shall accomplish it 's diurnal or daily Revolutions and by consequence it shall then lose 20 Seconds The Equations thus found for every particular day and a Table composed of them shall resemble that which is here inserted whose Use we now come to shew more particular in some Cases For Explanation take notice that the first Column contains the daies common to every Month the other 12 Columns that belong to the several Months themselves contain those Seconds of time that all natural daies are either longer or shorter than the mean day Note that in four parts of the Table are placed this Character ☉ which denotes the times wherein natural daies having before been longer than the mean day do then begin to be shorter or having before been shorter do then begin to grow more long Note also that those daies upon which this Character ☉ is affixed have no Equation they being equal in their length to the mean day as for the words inserted among the Columns they are at sight to inform you that the Equations in those parts of the Table are either more or else less than the mean day as the words themselves do fully express they also note that where the Equations are more there Clocks shall gain each day so much as the Equation belonging to it does then express but if the Equations are less they then shall lose and how much this gain or loss for every particular Months time shall amount to is by continual addition of the Equations belonging to each day summed up and the quantity of time it amounts to set down apart at the bottom of every Column Note also that since Clocks do either gain or lose during the whole number of daies included between those daies on whom this Character ☉ is affixed the whole quantity of
OF THE UNEQUALITY OF NATURAL TIME WITH ITS REASON and CAVSES TOGETHER WITH A TABLE OF THE TRUE AEQUATION OF NATVRAL DAYES Drawn up Chiefly for the Use of The GENTRY in Order to their more true Adjusting and right Managing of Pendulum Clocks and Watches By John Smith C. M. LONDON Printed for Joseph Watts at the Half-Moon in St. Paul's Church-yard 1686. A TABLE of Equations FOR REDUCING The Unequality of NATURAL DAYS TO A MEAN and EQUAL TIME Designed chiefly in order to the more true Adjusting and right Managing of Pendulum CLOCKS and WATCHES By JOHN SMITH C. M. Day Janua Sec Febru Sec. March Sec. April Sec. May. Sec. June Sec. July Sec. Aug. Sec. Sept. Sec. Octob. Sec. Nov. Sec. Dec. Sec. 1 Natural dayes longer than the mean day and Clocks gain 24 ☉ 0 Natural dayes shorter than the mean day and Clocks lose 17 Natural dayes shorter than the mean day and Clocks lose 17  3 Natural dayes longer than the mean day and Clocks gain 11 Natural dayes longer 7 Nat. dayes shorter than the mean and Clocks lose 9 Nat. dayes shorter than the mean and Clocks lose 20 Nat. days shorter Clocks lose 14 Natural dayes longer than the mean and Clocks gain 9 Natural dayes longer than the mean and Clocks gain 29 2 23 Natural dayes shorter than the mean day and Clocks lose 2 17 16 3 11 7 9 20 14 10 30 3 22 2 18 16 1 12 7 9 21 13 10 30 4 21 4 18 15 ☉ 0 12 6 11 21 13 10 30 5 20 4 18 15 ☉ 0 12 6 12 21 13 11 30 6 19 4 18 14 Nat. dayes longer than the mean day and Clocks gain 1 13 6 13 21 13 12 30 7 18 5 18 14 2 13 6 13 22 12 12 30 8 17 5 18 14 2 13 5 14 22 11 13 30 9 16 6 18 14 3 13 4 14 22 10 15 30 10 16 8 18 13 3 13 4 15 21 9 17 30 11 16 8 18 13 3 13 3 15 21 8 17 30 12 16 9 18 12 4 13 2 15 20 7 18 30 13 16 9 19 12 4 13 2 16 20 7 18 30 14 15 10 19 11 5 13 1 16 20 6 19 31 15 15 10 20 11 5 13 ☉ 0 17 20 6 20 31 16 14 11 20 10 6 12 ☉ 0 17 20 5 21 31 17 13 12 20 10 6 12 Nat. dayes shorter 1 17 20 4 22 31 18 12 13 20 10 6 11 2 18 20 3 23 30 19 11 13 20 10 6 11 3 18 19 3 23 30 20 11 13 19 10 7 11 4 19 19 2 24 30 21 10 14 19 9 8 11 4 19 19 ☉ 0 24 30 22 9 14 19 7 8 11 4 19 19 ☉ 0 24 30 23 8 15 19 7 9 10 4 19 19 ☉ 0 25 30 24 6 15 19 7 10 10 5 19 18 Longer 1 25 29 25 5 15 19 6 10 10 5 20 17 2 25 28 26 4 15 19 5 11 10 5 20 17 2 25 28 27 3 16 19 5 11 10 5 20 16 3 26 28 28 3 17 19 5 11 9 6 20 16 4 26 27 29 2  19 4 11 9 7 20 15 6 27 27 30 1 18 4 11 8 8 20 15 7 27 25 31 ☉ 0 17  11  9 20  8  24  Clocks gain this Month Clocks lose this Month Clocks lose this Month Clocks lose this Month Clocks gain this Month Clocks gain this Month Clocks lose this Month Clocks lose this Month Clocks lose this Month Clocks lose this Month Clocks gain this Month Clocks gain this Month Sum Min. Sec. Min. Sec. Min. Sec. Min. Sec. Min. Sec. Min. Sec. Min. Sec. Min. Sec. Min. Sec. Min. Sec. Min. Sec. Min. Sec. 6 26 4 29 9 37 5 16 2 47 5 43 0 6 8 23 9 41 2 20 9 38 15 9 This Table contains those Seconds of Time that each natural Day is either longer or shorter than the mean or equal day or such a one as contains in length the just time of 24 hours from which Mean Day the Natural ones differing almost continually in length this Table shall still give you the difference between them and by inspection only inform you what quantity of time each natural day is either more than 24 hours long or less Note That upon each particular day a Clock that is well adjusted to a mean or 24 hour day shall then either gain or lose just so much time as that natural day is either longer or shorter than the mean day Therefore if a Clock having been first set right to the Sun the first day of any particular Month shall either gain or lose in that whole Month so much in time as the whole sum of Equations for that Month amounts to which you shall find noted down at the bottom of every particular Column then is it well adjusted to the mean Equal or 24 hour Day but if it have not either got or lost so much as the whole sum of Equations for the whole time it has gone in do's amount to then must its motion be regulated as occasion requires by skrewing up the Bob to make it go faster in case it has not got enough or else letting the Bob down lower to make it go slower in case it hath not lost so much as the whole sum of Equations do amount to for the time it has gone in OF THE UNEQUALITY OF NATURAL TIME c. THE Vibrations of a long and weighty Pendulum although it be justly esteemed to be the most exact and steady of all Natural Motion yet is it not capable of regulating the Index of a Clock to such a pitch of Perfection as continually to point out the same time that shall be given by the Sun on the Lines of an exact and true Dial. The truth of which is sufficiently made evident by the most exact and critical Experiments For let all the moving parts of a Pendulum Clock be contriv'd with the greatest Skill and Judgment and then made up by the most cunning and curious Hand and after all this be adjusted by the utmost Care and Diligence of Man yet shall not the Motion of it correspond so continually with the Hours given by the Sun but that in some considerable quantity of Time you shall be sensible of gain or loss in the Motion of it The true Reason of which Variation proceeds not from any Defect that may be attributed to the Motion of the Pendulum of whose exactness we are by many curious Experiments sufficiently sensible but rather from an Unequality legible and easie enough to be discover'd in the diurnal Motions of the Sun Vulgarly for the most part the Sun is indeed accounted to be the Standard and Measure of all equal Time and Men generally esteem Natural Dayes to be all of one length as containing the just time of 24 Hours but upon a more exact and curious Scrutiny these vulgar Suppositions are found to be false For neither is the Sun's Motion found to be exact being in appearance to
time either got or lost does amount to the summs that follow viz. Between the 1st of February and the 4th of May the time that a well adjusted Clock shall lose amounts by continual adding the Equations together to about 19 Minutes 29 Seconds between this 4th of May and the 15th of July it shall gain about 9 Minutes 43 Seconds from the 15th of July to the 23d of October it shall lose 22 Minutes 9 Seconds from the 23d of October to the last of January it shall gain 31 Minutes 55 Seconds All this is to be understood of a well adjusted Clock set right to the Sun at the beginning of each time of either gaining or losing By this Table if you would adjust a Clock to a mean time which is the greatest exactness to which it's possible to be brought do thus First set it true to the Sun and note the day then let it's motion be continued without setting a new for about 30 or more daies Observe then the time that it has got or lost by the Sun then summ up the whole number of Seconds included in the Table between those two daies of first setting and last Observation allowing 60 Seconds to a Minute and if the gain or loss of your Clock be equal to the summ of time that it should have gained or lost by the Table then is it well adjusted but if it have not then must its motion be reduced to a more near agreement by shortning the Pendulum in case the Clock have gone too slow or letting the Bob down longer in case it have gone too fast Then set it anew and try it for about 30 daies more and then comparing its loss or gain with the summ of those Equations contained in the Table as before you did let the Bob be again rectified as the nature of it's motion requires and continue to do thus till you find its gain or loss exactly to correspond with the summ of time given by the Equations contained in the Table for the time that the Clock has gone When it is thus well adjusted to a mean time it will be so exact as that being set right at any time of the Year and so let go the whole Year about it shall come right with the same Dial by which it was set the same day Twelve-month but â—n all other parts of the Year it shall still differ from the same Dial. For Example If set right the first of February and so continued in Motion the whole Year about it shall continually be too slow the whole Year either more or less till the same day on which it was set The reason of this is plain enough for from the first of February to the 4th of May it shall continually lose to the quantity of 19 Minutes 29 Seconds then from the 4th of May to the 15th of July it shall gain but this gaining amounting to but about nine Minutes 43 Seconds it shall still be too slow by 9 Minutes 46 Seconds because its gaining now shall not be so much as it lost before by 9 Minutes 43 Seconds Then again from the 16th of July it shall lose afresh till the 21st of October which second loss amounting to about 22 Minutes 9 Seconds this added to the time that it was too slow on the last account shall amount to 31 Minutes 55 Seconds and so much it shall be too slow on the 21st of October from whence it shall gain afresh till the last of January to the quantity of 31 Minutes 55 Seconds which being equal to what it was before too slow shall cause it to come right to the same Dial with which it was set twelve Months before altho' it went too slow the whole Year beside Again let a Clock be set right the 23d of October it shall from thence gain time till the last of January and this gain shall amount to 31 Minutes 55 Seconds then from the first of February to the 4th of May it shall lose 19 Minutes 29 Seconds which being less than the 31 Minutes 55 Seconds which before it had got by about 12 Minutes 26 Seconds it shall still be too fast by 12 Minutes 26 Seconds Then from the 4th of May to the 15th of July it shall gain anew to the quantity of about 9 Minutes 43 Seconds which added to the time it was too fast before shall amount to 22 Minutes 9 Seconds and so much it shall be too fast on the 15th of July from which time till the 23d of October it shall lose this 22 Minutes 9 Seconds and by Consequence come right to the same Dial with which it was set twelve Months before Thus shall one and the same Clock with the same pitch of Motion go alwayes too slow if set at one time of the Year and always too fast if set at another time if it be let go the whole Year about Moreover if set at some other times and then continued in its Motion for a Year without setting anew it shall both gain and lose be sometimes too fast and sometimes too slow For if a well adjusted Pendulum be set right to the Sun the 4th of May by the 15th of July it shall be 9 Minutes 43 Seconds too fast From this 15th of July to the 23d of October having lost 22 Minutes 9 Seconds from which substracting the 9 Minutes 43 Seconds that it was before too fast there remains 12 Minutes 26 Seconds and so much it shall be too slow on the 23d of October from which day it shall begin to gain and continue so to do till the first of January by which time the Clock having got 31 Minutes 55 Seconds which amounting to about 19 Minutes 29 Seconds above what it was too slow on the 23d of October it shall by Consequence be now 19 Minutes 29 Seconds too fast from whence to the 4th of May it shall lose what now it is too fast and so come right to the same Dial with which it was before set Again set a Clock to the Sun the 15th of July and if it be well adjusted it shall by the 23d of October be 22 Minutes 9 Seconds too slow from whence to the last of January it being to gain 31 Minutes 55 Seconds it shall be then 9 Minutes 46 Second too fast from which time to the 4th of May it losing 19 Minutes 29 Seconds it shall then be 9 Minutes 29 Seconds too slow which time by the 15th of July shall again be got and so the Clock shall come right to the same Dial. Thus by this Table are these great Varieties discoverable in the Motion of the best adjusted Pendulum according to the different times of the Year that it is set in that the same Pendulum set right upon the first of February shall go always too slow till the same day twelve-month but if set right the 23d of October it shall the whole Year round be still too fast till the same day on which it was