C A L C U L AwT-l. % N S 0 N American Population,’ WITH A Table for eftimating the annual Increafe of Inhabitants in the Britift Colonies : The Manner of its Conftru&ion Explained { AND Its Ufe Illuflrated, b y EDWARD W.IGGLESWORTH, M. A, Hollis Profeilbr of Divinity at Cambridge. able ' toward Heaven, and tell theSTARs, if thou bo number them. So fhail thy Pofterity be. Benediction or; Abraham. Let there be no Strife, I pray thee, between us ; for we are Brethren. Is not the whole Coun¬ try before thee ? Abraham to his Kinfman Lot, BOSTON: Printed and Sold by John Boyle in Marlboro’-Street, T O THE HONORABLE THOMAS CUSHING, Efquire, S p e a k e r of the late H o v s e of REPRESENTATIVES Of the Province of the MASSACHUSETTS-BAY: AND One of their Delegates at the Continental CONGRESS, The following Calculations on American POPULATION, Publifhed under his Patronage, Are humbly infcribed. By his obliged Friend ,And obedient Servant, Tie AUTHOR. ’T' H E following Calculations were the Employment of fome leifure Hours, and defigned as a Recreation after Studies of a more ferious Nature. They were at firft undertaken for pri¬ vate Gratification. They are now made public, at the Recom¬ mendation of fome Triends, to 'whom they were communicated in Manufcript. tamhridge, Jan{ 25> i 775* CALCULATIONS G N AMERICAN POPULATION. F R O M a comparifon of the moft au¬ thentic eftimates of the inhabitants in the feveral Britifi colonies in America, taken at different times, it appears that the Britifh Americans have doubled their num¬ bers, in every period of twenty-five years from their firft plantation, A rapidity of popula¬ tion not to be parallaled in the annals of Eu¬ rope ! It has never been equalled fince the patriarchal ages. This rapid population of the Americans arifes, partly from the great acceflion of foreigners, but piincipally fr n m the natural increafe of the inhabitants. The reafons, why the Americans are more prolific than the Europeans, are, that they are lefs luxurious in their manner of living, and the means of fupporting a family can be more cafiljr ( 5 ) eafily obtained. For the lafl reafon the A- mericans are induced to marry earlier in life, and confequently their families of children are more numerous. And their temperance in diet renders them more healthy. The eafe of procuring fubfifience for a family is occafion- ed by the boundlefs traefs of uncultivated fo- refts, bordering on their plantations. For e- very new-married couple can, at fmall ex¬ pence, purchafe a freehold ; which by their induftry, will afford them and their children a comfortable fupport. Tilt, the wildernefs back of the old plan¬ tations fliall be filled with inhabitants, and brought under as good cultivation as the fettle- ments on the fea-coaft, it may be prefumed that the future population of thel'e colonies will be as rapid as the paft. By an eftimate made, at the late Coktinental Congress, of the number of perfons in eleven of the colonies, it appears thac there are 3,026678 fouls, inhabiting thofe colonies. Allowing that there are 526678 Haves, which allowance is probably larger than ought to be made, there are 2,500000 Britilh Americans, which may be reckoned as a Captial Stock, on which a calculation may be made of their future population. Should nothing intervene to re¬ tard ? 7 ) tard their increafe, the Americans will bg more numerous than their brethren in Bri¬ tain, in half a century from the prefent time : And in lefs than feventy years, their numbers will exceed thofe of the inhabitants of Greac- Britainand Ireland, taken colleftively.—Hap¬ py had it been for America, if its prefent con- teft with the parent ftate had been poftponed to the middle of the next century ! And more happy (till had it always flept in filcnce ! Britain and America might always have been fubfervient to each others profperity and hap- pinefs •, and have formed one of the mod po¬ pulous, potent, wealthy and happy kingdoms that ever exifted. Upon the fame principles that Mr. Ward' has conftru&ed a table to compute the com¬ pound intereft on any fum of money for the component parts of a year, a table may be formed to compute the number of inhabitants in any American colony for any given year.— Let i reprefent the prefent number of in¬ habitants, and a the number at the begin¬ ning of the next year. Then it will be as 22334 25. 1 : a :: a : a :: a : a : : a : a -f-f- to a That is, As the prefent number of inha¬ bitants is to the number of the fecond year, fo is that number to the number of the third year, and fo on in the fame proportion to the twenty-fifth year. Th^n The Operation* ,0858 ■+ e — oooei . 8581 ).0,000001188789 ( ,cc0033Szgae + *=_2i j 3 S 8l __ 35823 330689 _38 . '•5746 9 . 858268 7322000 + e 7066144. 8582762 25585600 85827649 842007 , !co ,772448^41 6 9SS 8753 r a 1,0281 4 « = .000013829 = «= 3,028113629 By this operation we find the value of « true to the tjth figure, and only too much by 2 in the 10th.—And by involving the value of a to its 25th power, we have the amount of unity to the 25th year; the index of the power, denoting the particular year. If there¬ fore the number, correfponding to any parti¬ cular power of a , be multiplied into the pre- fent number of inhabitants, either of any one of the Britifh American colonies, or of the whole continent, the produft will be the amount of inhabitants of that colony, or of the continent, for the year denoted by the index of the power of a. \ *0 ) The ratio of annual increafe may befouiid, and a table conftru&ed for eftimating the an¬ nual amount of the Americans, by a procefs lefs operofe and equally exadt. The pro¬ cefs is this: Take the logarithm of 2 —a from any table that will afford a fufficient num¬ ber of figures for the propofed calculation ; di¬ vide it by 25 the index of a, and the quotient will be the logarithm of a. This laft logari¬ thm being multiplied by 2, 3, 4, 5,&c. to 24 will give the logarithms of the intermediate powers of a ; the feveral multiplicators being the indices of their refpedtive powers. Find the natural number correfponding to each of thefe logarithms, and it wdl be the amount of unity from the iff to the 25th year inclufively. The following table was conftrudled, by afcertaining the ratio of the annual increafe of the Britilh Americans, according to the me¬ thod of folving adfefted equations. But in this method, where the quantities are furd, there will always be a defeB, or excefs, in the roots found. The error will increafe'at every invo¬ lution, and be proportionate to its power. In the procefs the value of a was taken true to .even figures, but was tooTittle by about ~V in the eight. When this number was involv¬ ed r ii ) cd to its 25th power, it difcovered itfelf, in the laft operation, to be,ooooi9lefs than juft5 and confequently that there was a proportio¬ nable defedt in each involution. The excefs or defefl arifing from the involution of a root greater or lefs than juft, may be corredted by the following proportion. As the value found Z S of a : to its excefs or defedl :: the value found of any other power: to the excefs or defett of that power. And the feveral defers being added to, or the feveral exceffes being fubftrac- ted from, the numbers firft found, will be the value of the feveral powers of a, fo near the truth as to anfwer all the purpofes of a calculation of this kind.-But the ftnall er¬ ror noticed above was corredted by taking the logarithm of 2 from Mr. Briggs 's table, and afcertaining the value of the feveral powers of a, in the method already noticed. By his ta¬ ble the annual amount of unity may be taken true to nine places of decimals, which is a greater degree of exactnefs than the propofed plan required. For this reafon eight decimals only were taken into the calculation. And therefore by multiplying the inhabitants by the numbers in the table, we have their amount for each year, without any error, where they are lefs than 10,coo 000 : A degree of exact - n.efs, fufficient for any calculation of this kird. At ( 12 ) At the Continental Congress held at Philadelphia the laft autumn, an eftimate was made, as has been noticed already, of the number of fouls in the provinces fpecified.— The publifher has not informed his readers •whether it includes Jlaves, (for to the difgrace of America slavery f ftill prevails here) as well as freemen. It is to be regreted that the publifher has not been more particular in his account of the eftimate. For if the num¬ bers in it are expreffive of freemen only, we have a foundation on which we may with fafety build a calculation. But if Jlaves are included Profefljr MAlar in fits curious off r-vat hits ok the difindion of ranks in focietyj having fpoken of ** the (lavery ellablilhcd in our colonies',” and hinted itt fume improvements which would render that ftate lefs intolerable, concludes with the following remark. “ Nothing can appear more aftonifhing than the little " attention that has hitherto been paid to any im- “ provements of this nature, after the good effeds of ** them have been fo fully illustrated in the “ cafe of the villains in Europe. -At the fame time it affords a curious fpe&acle to obferve, that the fame “ people- who talk in fo high a llrain of political liberty, “ and who conftder the privilge of impofing their overt “ taxes as one of the unalienable rights of mankind, i8 o99266, 7 [1,21419488. 1 8 1 » 24 < 838 ° 54 - 9 1,28342369. i° 1,31950791. n | 1,35660432. I 12 I 1 , 39474300 -j J 3 ~ J) 433955 2 14 1,47426921 . '5 1 > 5 ij 7 lb 56 . 16 1,55832916. ■ 7 1,60213975. 18 1,64718203. 19 1,69349062. '20' 1,74* IQ112- 21 1,79005014. It? 84037530. I 23 1,89211 5 ^ 9 * ]Z h9*53°!>fy- 2 r 2 , 00000000 . By this table the num¬ ber of inhabitants for any future year, the increafc of any particular year, and the increafe of any feries of years, may rea¬ dily be found. It will be bell to il- luftrate each of thefe cafes by a rule and example. Rule I. Deduct the prefent year 1775, from the year whofe inhabitants are re¬ quired, and if the differ¬ ence is lefs than 25, then the prefent number of in¬ habitants mull be multi¬ plied by the figures cor- refponding to the num¬ ber equal to fuch differ¬ ence, The produft will be the number required, and its decimal parts. Rejedl the eight right- hand figures, and the re¬ maining left-hand figures will be the number of inhabitants for the year' required. (' ts )\ Example I; What will be the number of inhabitants in the Maffachufetts-Bay A. D.1793, Deduct 1775 from 1793. The difference is 18, which being lefs than 25, the prefent' number of inhabitants muff be multiplied by- the figures in the table correfponding to 18.—* The inhabitants are 400000, which multiplied by 1,64718203, the produdt will be 658872,- 81200000. Cut off 8 figures on. the right- hand ; and the remaining figures on the left 658872 will be the number of inhabitants in the year 1793. But if the difference between the prefent. year, and that whofe inhabitants are required be greater than 25, the following procefs mull be obferved. Rule II,' Divide the difference by 25, then double the prefent number of inhabitants as many times as there are units in the quotient. Multiply the product of the number fo doub¬ led by the figures in the table correfpondenc to the Remainder- Rejedl from the product ? vs y ITi'e eight right-hand figures, and the figured pn the left will be the number fought. Example IL What will be the number of Britifh A- mericans, A. 1890, allowing them to be no more than 1500000 at prefent: From 1890, fubftradt 1775, the difference is 115, which being more than 25 mull be divided by it. The quotient of 115 divided by 25 is '4, and the remainder 15. Double 1500000 four times, which will give 24,000000. This fum being multiplied by the figures in the table, anfwering to 15, being 1,51571656, will give 3637719744000 000. Rejeft the 8 figures on the right-hand, and the others will be the number of Britifh Americans, A. D. 1890. viz. 36,377197 perfons. The increafe of inhabitants in any given year may be determined in the following method. Rule III. Find the number of inhabitants at the be- giningofthe year by the 1 ft or 2d rule. Mul¬ tiply the number fo found by the decimals in the firft line of figures in the table. Rejeft 8 figures on the right-hand, and thole that are left will give the anfwer, Fx- C 11 5 £ X A M P L E Ilf. What will be the increafe of the Britifh Americans, A. D. 189Q ? By the laft example their number at the be¬ ginning of that year is found to be 36,377197. Multiply this number by 0,02811382, the de¬ cimals in the firft line of figures in the table, and the product will be, 1022701,96856254. Rejeft the decimals on the right-hand, and the integers will be the anfwer to the queftion, viz. 1,022701 is the addition that will be made to the Britifh Americans in the year 1890; fuppofing them to be no more than one million and an half at prefent. The increafe of inhabitants for any feries of years may be afcertained in the manner following. Rule IV, Find the number of inhabitants for the firft and laft years in the feries, dedudt the lefs from the greater number, and the remainder will be the number fought; Example IV. How many will the inhabitants of the Maf- lachufetts colony increafe by the year 1793. C Their { H ) Their number in 1793 will be — 658872,' Their prefent number is — :-400000. 25S872 will be the increafc at that time. By the fame table may likewife be deter¬ mined, in a procefs fomething different from the preceeding, both the number of inhabi¬ tants in any year pad •, and alfo what propor¬ tion of the prefent inhabitants are descended from original planters, where the number of them can be afcertained, and what from per- fons who left Europe after the firft implanta¬ tion of the particular colony, whofe inhabitants are the fubjeft of difquifition. The number of inhabitants for any year pad may be determined in one of the follow¬ ing methods. Rule V. Take the difference between the prefent year, and that whofe inhabitants are required, and if it be lefs than 25, deduct from it 25. And one half the prefent inhabitants of the colony mud be multiplied by the num¬ bers in the table correfponding to the remain¬ der. The integers in the produft will give the number of inhabitants for the year required. Ex* ( *9 > Example V. What was the number of inhabitants, A. D. 1758, in the Maflachufetts ? The difference of years is 17, this deduced from 25, leaves 8. One half of the prefent inhabitants is 200,000. This number mul¬ tiplied by 1,24833054, the figures in the table correfponding to 8, the remainder, will give 249666,10800000. Rejefting the decimals, the anfwer is, the Maffachufetts inhabitants were A. D. 1758, 249666 perfons. But if the difference of years be greater than 25, the number of inhabitants mu ft be determined- by R u l e VI. Divide the difference of years by 25, and fubftracl the remainder from 25. Take one half of the prefent inhabitants and divide them fucceffively by 2, as many times as there are units in the quotient. The inhabi¬ tants thus depreffcd, being multiplied by the figures in the table, anfwering to the dif¬ ference between the remainder and 25, will be the anfwer required, after the decimals are rejected. Example VI. What was the number of Americans, 1710? The ( 20 ) The difference of years is 65. This num¬ ber divided by 25 gives 2 for the quotient, ?nd 15 for the remainder. One half of the prefent number of Americans is, 1,250000. This number depreffed, by being twice divi¬ ded by 2, will give 312,500, which multi¬ plied by the figures in the table anfwering to 10, (which is the difference between 25 and 15, the remainder,) 1,31950.791 will make 412346,22187500. Rejedt the decimals, which will give the anfwer. The number of Americans A. D. 1710, was 412346. Should the proportion be fought, which the po.fterity of the firft fettlers of any colony bear to the whole inhabitants of it at prefent, it may be afeertained in the following method, fuppofing them to double by natural increafe in twenty-five years. Rule VII. Deduct the year of the fettlement of the polony from the prefent year. Divide the dif¬ ference of years by 2. Then multiply the origi¬ nal fettlers by 2, as many times as there are units in the quotient. The numher thus encreafed, multiply hy the figures in the table, anfwer¬ ing to the remainder. The produdl, when the decimals are rejedted, will give the prefent number of the pofterity of the original fet¬ tlers. And the proportion \yhich they bear tQ ( 21 ) to the whole, will be as that number is to this number of the whole. E P A M p l e VII. What proportion do the pofterity of th$ original planters of the Maffachufptts, bear to its' prefent inhabitants ? Could the number, pf original, fettlers be afcertained, the queftion might be determined with precifion. The rule however may. be il- luflrated by taking any number at pleafgre, inftead of the true number of original fet¬ tlers.-Let us then fuppofe the firft fettlers amounted to 5000. Massachusetts was planted A. D. i6^0.f The difference between that and the prefent year is 145. This number divided by 25, gives. 5 for the quotient, and 20 fur the remain¬ der. 5000 redoubled 5 times makes 160000. This number multiplied by 1,74110112, the figures in the table correfponding to 20, the remainder, will give 278576,17920000. The defendants, then, from the original fettlers are, on the fuppcfition made, 278576; and they are in proportion to the whole inhabitants, as 278576 is to 400000. By doubling the prefent number of Ame¬ ricans nine times, the number of them will be found f Gov. Winthbop arrived at Salem June 12, 1650. The fettlement at Plimoutb was begun De- ember 20, 1620, ” f 22 ) Found for every twenty fifth year to the end of the twentieth century. And each of thefe duplications,being multiplied by thefeveral lines of figures in the table, will give the amount of the Americans in the intermediate years.— Their prefent number being eftimated at two millions and an half, their duplications will be as follows, 'A.I8oo —5 Millions. A. 1925— ifioMillions. 1825—10 Do. 1950—320 Do. 1850—20 Do. 1975—640 Do. 1875—40 Do. 2000—1280 Do. 1900—80 Do. A careful attention to the preceeding rules and examples, will be fufficient, without any more illuftrations, to lead to the know- lege of the various purpofes to which the ta¬ ble may be applied. When we look back to the ftateof the co¬ lonies at the middle of the laft century, and compare it with the prefent, we are furprifed to find that our anceftors, amidft all the diffi¬ culties they had to encounter, have been able in fo (hort a period to put a face entirely new on all the country extended from Nova-Scotia to Georgia, by changing the forefi into a fruitful field i that they have opened fuch an extenfive commerce, as is carried on from A- merica ; and that by their cultivation of the liberal I 23 5 liberal arts, .they- have lefc a pofterity of tw« millions and a half, § enjoying all the necefla- ries, and molt of the elegancies of life. A rapidity of population and improvement, that fills the mind with admiration ! But to anti¬ cipate their population and improvements, at theclofe of the twentieth century,overwhelms the mind with aftonilhment ! At that time, fhould their future population be as rapid as their pad, the Americans would amount to On^ thousand two-hundred and eighty, millions ! The continent extending back to the fouth-fea, affording them new plantations; and the diverfity of climates and foils, invi- § Five millions, according to the eftimate of—Dr... Price “ and our bell modern calculators, is neared to “ the true number of the people of England. The “ males between fixteen and fifty-fix in five millions “ are 1,250,000, or a fourth part of the whole.” “ In North-Britain the number of fouls is about •• 1,500,000. The males between fixteen and fifty- “ fix are 300,000.” Political ai/juifiiiom Vo!. I. page 36, 46.-If the Do&or’s proportion is juft, the number of males in America between fixteen and fifty- fix is 625,000.--The increafe of Americans the prefent year, is 70,284. One fourth part of them are ^ males between fixteen and fifty-fix. There are then 17,571, effeftive men added to the Americans the pre¬ fent year. Lord Barrington moved in the Honfe of Commons, that 17,542 effective men be employed for the land fervice this year. I H > tih'gf the inCrodu&ion of all the various produce tiotis of the other quarters of the world. Should afyftem of policy be adopted to conciliate the affedlions of the Americans, What an amazing fource 6f commerce will be opened to Great Britain ? To What limits will the demand for Britilh manufactures be ex¬ tended, while the Americans ate bufily em¬ ployed in fubduing the wilderricfs ? And fHouId Britain continue to encotifage the cul¬ tivation of thefe materials of r commerce, for which flie has been dependent on foreign nations, to what a date of independence may fhe arife ? That the prefent unhappy conteft between Britain and America, may be amicably termi¬ nated ; that the extent of parliamentary autho¬ rity may be afcertairied; and that the rights of America may be fixed oh an immoveable foundation,are the earned defires of every true friend of Britain and America. Should this ever be the cafe, fuch an union of intereft and affe&ion would fucceed, as would render them the envy of Europe and the glory of the World. THE END.