The welspring of sciences, which teacheth the perfecte worke and practise of arithmeticke both in vvhole numbers & fractions, with such easie and compendious instruction into the saide art, as hath not heretofore been by any set out nor laboured, : Beautified vvith most necessary rules and questions, not onely profitable for marchauntes, but also for all artificers, as in the table doth plainely appere..

¶Of progression the vi. Chapiter.

PRogression arithmetical,* 1.1 is a briefe & spedy assembling or adding together of diuers fi∣gures or nombres, euery one surmounting the other cōtinually by equall difference: as 1.2.3.4.5. &c. here the differēce, from the first to the secōd is but of 1. and so do al the other, euery one excede another by 1. still to thend. Like waies. Here .2.4.6.8. &c. do pro∣cede by the difference of 2. also 3.6.9. 12. &c. doe euery one differ from other by 3. and so may these noumbres con∣tinue. Infinitelie after this order, in adding vnto the thirde noumbre, the quantitie wherein the seconde dothe differ from the fyrste: Lyke wayes

addinge the same difference vnto the fowerth noumbre, also to the fyfte, and so vnto all the other. As .1.4. the difference of the seconde to the fyrst is 3. adde 3. vnto 4: and they are 7. for the thirde noumbre: Then adde 3. vn∣to 7: and thei make 10. for the fowrth noumbre, and so of all other.

Then if you will adde quickely the noumbre of any progession, you shall dooe thus, first tel howe many noum∣bres there are, and wryte their somme downe by it selfe, as in this example, 2. 5. 8. 11. 14. where the noumbres are 5. as you maye sée, therefore you must sette downe 5. in a place alone, [unspec 5] as I haue done here in the margent. Then shall you adde the first noum∣bre and the last together, whiche in this exaumple are .14. and 2. and they make .16. take halfe thereof whiche is .8. and multiplie it by the 5. whiche I nooted in the margente for the noumbre of the places, and the som∣me whiche amounteth of that multi∣plicacion,

is the iust somme of al those figures added together, as in this exā∣ple: 8. multiplied by .5. doe make .40. and that is the somme of all the figu∣res. An other example of parcels that are euen, as thus .1.2.3.4.5.6. in this exāple you must likewaies note doun the nomber of the places, as before is taught, and thā adde together the last nomber and the first. And the somme, whiche cometh of that addicion, shall you multiplie by halfe the nomber of the places, whiche before are noted, and that, whiche resulteth of the same multiplicacion, is the wholesomme of all those figures, as in this former ex∣ample, where the nomber of the pla∣ces is .6. I note the .6. a part, and then [unspec 6] I adde .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 moche amoū∣teth all those figures, added together.

Progression Geometricall is,* 1.2 when

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 fift the fowerth. &c. As .2.4.8.16.32, 64: here the proporcion is double.

Likewaies .3.9.27.81.243. are in triple proporcion.

And .2.8.32.128.512. are in propor∣cion quadruple.

That is to saie, 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 Arithmeticalle, differeth from Progression Geometricalle for that, that in the Arithmeticalle.

The excesse is onelie in quantitie, but in the Geometricalle, 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 multiplicacion, 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 quocient 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 shall 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:

which is the total somme of the nom∣bers of that progression. Like wise .4. 16.64 256.1024. whiche are in pro∣porcion quadruple: therfore multiplie 1024. by .4. and thereof cometh 4096 from the whiche abate the firste nom∣ber .4. and there resteth .4092: The whiche you must diuide by .3. and you shall finde in your quotiente .1364. whiche is the total somme of that pro∣gression, and this shalbe sufficient for progression.

Notes

• * 1.1

  Progression arithmetical.
• * 1.2

  Progession, geometricall.