1double, and the quantity increaſing nonuple, the height increa
ſeth triple; ſo that, by adding to units all the odde numbers, ac
cording to their Series, the heights increaſe according to the na
tural progreſſion of all the numbers, from units. As for exam
ple, there paſſing thorow a Regulator ſuch a certain quantity of
Water in one time; adding three of thoſe meaſures, the quick
height is two of thoſe parts, which at firſt was one; and con
tinuing to adde five of thoſe ſaid meaſures, the height is three of
thoſe parts which at firſt were one; and thus adding ſeven, and
then nine, and then 11. and then 13, &c. the heights ſhall be 4.
then 5, then 6. then 7, &c. And for the greater facility of the
Work, we have deſcribed the following Table, of which we will
declare the uſe: The Table is divided into three Series or Pro
greſſions of Numbers: the firſt Series containeth all the Num
bers in the Natural Progreſſion, beginning at a Unit, and is called
the Series of the Heights; the ſecond containeth all the odde
numbers, beginning at an unit, and is called the Series of the
Additions: the third containeth all the ſquare numbers, begin
ning at an unit, and is called the Series of Quantity.
Heights.1234567891011Additions.13579111315171921Quantities.149162536496481100121ſeth triple; ſo that, by adding to units all the odde numbers, ac
cording to their Series, the heights increaſe according to the na
tural progreſſion of all the numbers, from units. As for exam
ple, there paſſing thorow a Regulator ſuch a certain quantity of
Water in one time; adding three of thoſe meaſures, the quick
height is two of thoſe parts, which at firſt was one; and con
tinuing to adde five of thoſe ſaid meaſures, the height is three of
thoſe parts which at firſt were one; and thus adding ſeven, and
then nine, and then 11. and then 13, &c. the heights ſhall be 4.
then 5, then 6. then 7, &c. And for the greater facility of the
Work, we have deſcribed the following Table, of which we will
declare the uſe: The Table is divided into three Series or Pro
greſſions of Numbers: the firſt Series containeth all the Num
bers in the Natural Progreſſion, beginning at a Unit, and is called
the Series of the Heights; the ſecond containeth all the odde
numbers, beginning at an unit, and is called the Series of the
Additions: the third containeth all the ſquare numbers, begin
ning at an unit, and is called the Series of Quantity.
Firſt, if we ſuppoſe the whole quick height of a River of Run
ning Water to be divided into any number of equal parts, at
pleaſure, and would abate the ſame one fift, by means of a divi
ſron; let there be found in the Table in the Series of heights the
number 5. the denominator of the part which the River is to a
bate, and take the number that is immediately under it in the
row of Additions, which is 9. which let be ſubſtracted from the
number 25. placed underneath the ſame in the row of Quanti
ties, the remainder 16. ſignifieth that of the 25. parts of Water
that ran in the River, whilſt it was 5 meaſures high, there do
onely run 16. parts; ſo that to make it abate 1/5 it is neceſſary to
take 9/25 from the Water that the whole River did carry; ſo that
with ſubſtracting ſomewhat more than one third of the Water of
the River, it is abated but only one fift.
ning Water to be divided into any number of equal parts, at
pleaſure, and would abate the ſame one fift, by means of a divi
ſron; let there be found in the Table in the Series of heights the
number 5. the denominator of the part which the River is to a
bate, and take the number that is immediately under it in the
row of Additions, which is 9. which let be ſubſtracted from the
number 25. placed underneath the ſame in the row of Quanti
ties, the remainder 16. ſignifieth that of the 25. parts of Water
that ran in the River, whilſt it was 5 meaſures high, there do
onely run 16. parts; ſo that to make it abate 1/5 it is neceſſary to
take 9/25 from the Water that the whole River did carry; ſo that
with ſubſtracting ſomewhat more than one third of the Water of
the River, it is abated but only one fift.