Arithmetic, Part 1 · Section 22
Geometrical Progreſsion.
I ſhall not be ſo large in this as in the former, becauſe theſe things are of little uſe to the Arithmetician, except where a number is to be many times doubled, tripled, or the like, which cannot be ſo eaſily abridged here, as in the other, becauſe there the laſt number ariſing of many additions of the exceſſe to 1, was eaſily found by one multiplication: but here the laſt number being made by many Multiplications of the exceſſe, is therefore many times harder then the other
The varieties here ſhall be but two.
1 The common exceſſe and number of places being given, to find the laſt number.
2 The exceſſe, and laſt number being given, to find the total ſumme.
The firſt of theſe may thus be found. Let the numbers be 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, the exceeſſe is 2, the places 10, find out the fifth number (which is eaſily done, for any one may reckon ſo farre by heart,) that is here 16, and multiply 16 by 16, it produceth 256, which is the ninth number, laſtly, multiply 256 by the exceſſe 2, thence ariſeth 512, the number deſired.
So if the places had been more, as 72, having found the 9 number 256, multiply it by 256, thence comes 65536 for the 17th. number, which multiplied by the exceſſe 2, gives 131072 for the 18th. place, which multiplied by 131072, gives 17179869184, for the 35 place; and that multiplied againe by the exceſſe 2, gives 34359738368, for the 36 place, that multiplied by 34359738368, then the product will be 1180591620717411327|424, for the 71 place, which laſtly, multiplied by the exceſſe, gives 2361183241434822654848, for the 72 place, which is the laſt number of the progreſſion required to be found.
Perhaps this may ſeem ſomewhat tedious but where things cannot be performed without labour, the Reader muſt content himſelf with ſuch Rules as make it leſſe; for it is certaine that this way is much ſhorter then to have multiplied ſtill by the exceſſe 71 times, which elſe he muſt have done.
All this notwithſtanding, he is not bound to uſe the ſame numbers, much leſſe in other queſtions where the number of places is not the ſame, but whereas I began from the place 9, he may begin at 8, 10, or 12, or where he pleaſes; ſo as he remembers ſtill where he is; for this is general, if the number belonging to any place whatſoever, be multiplyed by it ſelf, the product ſhall be the number belonging to twice ſo many places want one place.
Now for the ſecond thing, which is to find the ſumme of all the numbers.
THE RULE.
From the laſt number take the firſt, and divide the remaine by the exceſſe want 1, then multiply the quotient by the exceſſe; and to the product adde the firſt number, the ſumme of them is equal to the ſumme of all the numbers.
So if from the laſt number, or 72 place be taken 1 remain is 2361183241434 22654847 which ſhould be divided by 1, (that is the exceſſe want 1 for the exceſſe is but 2) but becauſe 1 neither multiplies nor divides that labour is ſaved; Now multiply this remain by the exceſſe, the product is 4722366482869645309696, to which adding the firſt number 1, by making the figure 6 next the right hand to be 7, you have the total ſumme of all the 72 numbers.
A Queſtion reſolved by Geometrical Progreſſion.
A Londoner ſojourning in a Countrey Market Towne, in Winter, made himſelfe a new freez, Suit and Coat, on which were ſet 6 dozen of buttons of Silke and Silver; A Baker being in his company liked it ſo well he would buy it of him; the Citizen conſented to let him have it, paying for the firſt button a ſingle barley corn, for the ſecond 2, for the third 4, and ſo on doubling to the laſt.
The bargaine was liked on both parts for the preſent, but ſhortly after revoked, for it could not be performed, and no man can be holden to an impoſſibility.
But why this could not be performed: may be judged; firſt, by inquiring the worth of ſo much barley in Money: And ſecondly, the weight of it; and how it ſhould be removed.
1 For the firſt, allowing 10000 corns to a pinte (which is more then enough) then 5120000 corns make a quarter; and yet (for ſhortning the diviſion) we will allow 10000000 corns to a quarter; by which dividing the whole number of corns (which is done by cutting off the firſt 7 figures toward the right hand,) the quotient will be 472239648286964, and ſo many whole quarters there are omitting the remaine, as in this caſe not conſiderable.
Now allowing barley were to be ſold at 15 d. the buſhel (which is cheap,) it is ſo many Angels: and therefore dividing by 2 it is 2361183241414|3482 pounds ſterling: which is in words, Two hundred thirty ſix millions of millions, one hundred and eighteen thouſand, three hundred twenty four millions: one hundred forty three thouſand; four hundred eighty two pounds, whick I take to be too much for any Tradeſ-man to get or keep.
And reckoning land for ever at twenty yeares purchaſe, if this ſumme of pounds be divided by 20, the quotient is the yearly rent of 118059|16207174 pounds.
And this divided again by 365 (the number of dayes in a year) the quotient 32344975918, that is above thirty two thouſands of millions a day for ever. So great a vanity may be concluded on for want of a little premeditation.
2 Now, ſecondly, for the weight of it, if we put 8 Buſhels to weigh 2 hundred pound weight (for ſure it doth weigh more) then the whole number of quarters multiplied by 2, gives the weight of all to be 944473296573928 hundred weight, and if this be divided by 20, (which is but cutting off one figure toward the right hand, and dividing the reſt by 2) or which is all one, cut off one figure from the number of quarters, the quotient 47223664828696, is ſo many tuns. And therefore it will require 47223664828 ſhips of 1000 a piece to carry it: And conſequently if every Nation in the World had above 10000 ſuch ſhips, yet there muſt be above four millions of ſuch Nations: which I ſuppoſe are not to be found in this World.
And here I will leave this, having uſed this long example, (which though it require more labour as all great examples do, yet the ſame skill will do it, as if the places had been fewer) that the Reader being throughly exerciſed thereby, may the eaſier leap over others which are ſhorter.