The vvell spryng of sciences whiche teacheth the perfecte woorke and practise of arithmeticke, bothe in whole nombers and fractions, with suche easie and compendious instruction into the said arte, as hath not heretofore been by any sette out nor laboured. Beautified with moste necessary rules and questions, not onely profitable for marchauntes, but also for all artificers, as in the table doeth partlie appere: set forthe by Humfrey Baker citezeine of Lo[n]don.

¶ Of progression, the vi. Chapiter.

PRogression Arithmeticall, * 1.1 is a brief and spedie assem∣blyng, or addyng together of diuers figures or num∣bers, euery one surmountyng thother continually, by equall difference: as 1. 2. 3. 4. 5. &c. here the diffrence, from the first to the seconde is but of 1. and so do al the other, euery one excede an other by. 1. still to thende. Like waies. Here 2. 4. 6. 8. &c. doe proceade by the diffe∣rence of. 2. also. 3. 6. 9. 12. &c. doe euery one differ from other by. 3. and so may these nūters continue. Infinitty after this order, in addyng vnto the thirds number, the quantitie wherein the se∣cōde doth differ frō the first: like wates

addyng the same difference vnto the fowerth number, also to the fifte, and so vnto all the other. As. 14. the diffe∣rence of the seconde to the firste is. 3: adde. 3. vnto. 4: and thei are. 7. for the thirde number: Then adde. 3. vnto. 7: and thei make. 10. for the fowerth nū∣ber, and so of all other.

Then if you will adde quickly the number of any progression, you shall doe thus, firste tell how many num∣bers there are, and write their somme doune by it self, as in this example, 2. 5. 8. 11. 14. where the numbers are 5 as you maie see, therefore you muste sette doune. 5. in a place alone, as I * 1.2 haue dooen here in the margent.

Then shall you adde the first num∣ber, and the laste together, whiche in this example are. 14. and. 2. and thei make. 16. take halfe thereof, whiche is. 8. and multiplie it by the. 5. whi∣che I noted in the margente, for the nomber of the places, and the somme whiche amounteth of that multy∣plication,

is the iuste somme of all those figures added together, as in this example: 8. multiplied by. 5. dooe make. 40. and that is the somme of al the figures.

An other example of parcelles that are euen, as thus. 1. 2. 3. 4. 5. 6. in this exāple you must likewaies note doun the number of the places, as before is taught, and thē adde together the last nomber and the first. And the somme whiche commeth of that additiō, shall you multiplie by halfe the nomber of the places, whiche before are noted, and that, whiche resulteth of the same multiplication, is the whole somme of al those figures, as in this former exā∣ple, where the nomber of the places is 6. I note the 6. apart, and then I adde * 1.3 6. and 1. together, whiche are the laste and firste nombers, and thei make. 7. the whiche I multiplie by. 3. whiche is halfe the nomber of places, and thei make. 21. and so muche amounteth all those figures added together.

Progression Geometricall is, when * 1.4 the second nomber containeth the first in any proporcion. 2. 3. or. 4. times and so forthe. And in like proporcion shall the thirde nomber contain the second, and the fowerth, the third, and the first the fowerth. &c. As. 2. 4. 8. 16. 32. 64. here the proportion is double.

Likewaies. 3. 9. 27. 81. 243. are in triple proportion.

And. 2. 8. 32. 128. 512. are in propor∣cion quadruple.

That is to saye, in the firste exam∣ple, where the proporcion is double, euery nomber containeth the other. 2. tymes. In the seconde example of tri∣ple proporcion, the noumbers exceade eche other thre times. And in the third example, the nombers exceade eche o∣ther fower times, and thus you se that progression Arthmeticalle, differeth from Progression Geometricalle for that, that in y^• Arithmeticall. The ex∣cesse is only in quātitie, but in the Geo¦metricall, the excesse is in proporcion.

Nowe if you will easelie finde the somme of any soche nombers, you shal dooe thus, consider by what noumber thei be multiplied, whether by. 2. 3. 4. 5. or any other, and by the same nom∣ber you must multiply the last somme in the progression. And from the pro∣ducte of the same multiplication, you shall abate the first nomber of the pro∣gression. And that whiche remaineth of the saied multiplicacion, you shall diuide by. 1. lesse then was the nom∣ber, by the which I did multiplie. And the quotient shall shew you the sōme of all the nombers in any Progressi∣on. As in this exaumple. 5. 15. 45. 135. 405. whiche are in triple proporcion: now muste you multiplie. 405. by. 3. and thei are. 1215. from the which you shal abate the first nomber of the pro∣gression, whiche is. 5. and there resteth 1210. the whiche you shall diuide by the nōber lesse by. 1. then by the which you did multiplie, that is to saie, by. 2. and you shall finde in the quociēt 605.

whiche is the totall somme of the nō∣bers of that progression. Likewise. 4. 16. 64. 256. 1024. whiche are in pro∣portion quadruple: therfore multiplie 1024. by. 4. and thereof commeth 4096. from the whiche abate the first nomber. 4. and there resteth. 4092: the whiche you must deuide by. 3. and you shall finde in your quotient. 1364 whiche is the total somme of that pro∣gression, and this shalbe sufficient for progression.

Notes

• * 1.1

  Progression Arithmeti∣calle.
• * 1.2

  5.
• * 1.3

  6.
• * 1.4

  Progression. geometricall.