Of the mensuration of running waters an excellent piece written in Italian by Don Benedetto Castelli ... ; Englished from the third and best edition ; with the addition of a second book not before extant / by Thomas Salusbury.

OF THE MENSURATION OF Running Waters.

SUPPOSITION I.

LEt it be supposed, that the banks of the Rivers of which we speak be erected perpendicular to the plane of the up∣per superficies of the River.

SUPPOSITION II.

WE suppose that the plane of the bottome of the River, of which we speak is at right angles with the banks.

SUPPOSITION III.

IT is to be supposed, that we speak of Rivers, when they are at ebbe, in that state of shallownesse, or at flowing in that state of deepnesse, and not in their transition from the ebbe to the flowing, or from the flowing to the ebbe.

Declaration of Termes.

FIRST.

IF a River shall be cut by a Plane at right angles to the surface of the water of the River, and to the banks of the River, that same dividing Plane we call the Section of the River; and this Section, by the Suppositions above, shall be a right angled Parallelogram.

SECOND.

WE call those Sections equally Swift, by which the water runs with equal velocity; and more swift and less swift that Section of another, by which the water runs with greater or lesse velocity.

AXIOME I.

SEctions equal, and equally swift, discharge equal quantities of Water in equal times.

AXIOME II.

SEctions equally swift, and that discharge equal quantity of Water, in equal time, shall be equal.

AXIOME III.

SEctions equal, and that discharge equal quantities of Water in equal times, shall be equally swift.

AXIOME IV.

WHen Sections are unequal, but equally swift, the quanti∣ty of the Water that passeth through the first Section, shall have the same proportion to the quantity that pas∣seth through the Second, that the first Section hath to the second Section. Which is manifest, because the velocity being the same, the difference of the Water that passeth shall be according to the difference of the Sections.

AXIOME V.

IF the Sections shall be equal, and of unequal velocity, the quantity of the Water that passeth through the first, shall have the same proportion to that which passeth through the second, that the velocity of the first Section, shall have to the velocity of the second Section. Which also is manifest, because the Sections being equal, the difference of the Water which passeth, dependeth on the velocity.

PETITION.

A Section of a River being given, we may suppose another equal to the given, of different breadth, heigth, and ve∣locity.

PROPOSITION I.

The Sections of the same River discharge equal quan∣tities of Water in equal times, although the Secti∣ons themselves he unequal.

LEt the two Sections be A and B, in the River C, running from A, towards B; I say, that they discharge equal quan∣tity of Water in equal times; for if greater quantity of Wa∣ter should pass through A, than passeth through B, it would

[illustration]

follow that the Water in the intermediate space of the River C, would increase continually, which is manifestly false, but if more Water should issue through the Section B, than entreth at the Section A, the Water in the intermediate space C, would grow continually less, and alwaies ebb, which is likewise false; therefore the quantity of Water that passeth through the Secti∣on B, is equal to the quantity of Water which passeth through the Section A, and therefore the Sections of the same River dis∣charge, &c. Which was to be demonstrated.

PROPOSITION II.

In two Sections of Rivers, the quantity of the Water which passeth by one Section, is to that which pas∣seth by the second, in a Proportion compounded of the proportions of the first Section to the second, and of the velocitie through the first, to the velocitie of the second.

LEt A, and B be two Sections of a River; I say, that the quantity of Water which passeth through A, is to that which passeth through B, in a proportion compounded of the pro∣portions of the first Section A, to the Section B; and of the velo∣city through A, to the velocity through B: Let a Section be

supposed equal to the Section A, in magnitude; but of velocity equal to the Section B, and let it be G; and as the Section A is

[illustration]

to the Section B, so let the line F be to the line D; and as the velocity A, is to the velocity by B, so let the line D be to the line R: Therefore the Water which passeth thorow A, shall be to that which passeth through G (in regard the Sections A and G are of equal bigness, but of 〈◊〉〈◊〉 equal velocity) as the velocity through A, to the velocity through G; But as the velocity through A, is to the velocity through G, so is the velocity through A, to the velocity through B; namely, as the line D to the line R: therefore the quantity of the Water which passe the through A, shall be to the quantity which passeth through G, as the line D is to the line R; but the quantity which passeth through G, is to that which passeth through B, (in regard the Sections G, and B, are equally swift) as the Section G to the Se∣ction B, that is, as the Section A, to the Section B; that is, as the line F to the line D: Therefore by the equal and perturbed proportionality, the quantity of the Water which passeth through A, hath the same proportion to that which passeth through B, that the line F hath to the line R; but F to R, hath a proportion compounded of the proportions of F to D, and of D to R; that is, of the Section A to the Section B; and of the velocity through A, to the velocity through B▪ Therefore also the quantity of Water which passeth through the Section A, shall have a propor∣tion to that which passeth through the Section B, compounded of the proportions of the Section A, to the Section B, and of the velocity through A, to the velocity through B: And therefore in two Sections of Rivers, the quantity of Water which passeth by the first, &c. which was to be demonstrated.

COROLLARIE.

THe same followeth, though the quantity of the Water which passeth through the Section A, be equal to the quantity of Water which passeth through the Section B, as is manifest by the same demonstration.

PROPOSITION III.

In two Sections unequal, through which pass equal quantities of Water in equal times, the Sections have to one another, reciprocal proportion to their velocitie.

LEt the two unequal Sections, by which pass equal quantities of Water in equal times be A, the greater; and B, the lesser: I say, that the Section A, shall have the same Proportion to the Section B, that reciprocally the velocity through B, hath to the velocity through A; for supposing that as the Water that passeth through A, is to that which passeth through B, so is the

[illustration]

line E to the line F: therefore the quantity of water which pas∣seth through A, being equal to that which passeth through B, the line E shall also be equal to the line F: Supposing moreover, That as the Section A, is to the Section B, so is the line F, to the line G; and because the quantity of water which passeth through the Section A, is to that which passeth through the Section B, in a proportion composed of the proportions of the Section A, to the Section B, and of the velocity through A, to the velocity through B; therefore the line E, shall be the line to F, in a proportion compounded of the same proportions; namely, of the proportion of the Section A, to the Section B, and of the ve∣locity through A, to the velocity through B; but the line E, hath to the line G, the proportion of the Section A, to the Section B, therefore the proportion remaining of the line G, to the line F, shall be the proportion of the velocity through A, to the velocity through B; therefore also the line G, shall be to the line E, as the velocity by A, to the velocity by B: And conversly, the ve∣locity through B, shall be to the velocity through A, as the line E, to the line G; that is to say, as the Section A, to the Section B, and therefore in two Sections, &c. which was to be demonstrated.

COROLLARIE.

HEnce it is manifest, that Sections of the same River (which are no other than the vulgar measures of the River) have betwixt themselves reciprocal proportions to their veloci∣ties; for in the first Proposition we have demonstrated that the Sections of the same River, discharge equal quantities of Water in equal times; therefore, by what hath now been demonstrated the Sections of the same River shall have reciprocal proportion to their velocities; And therefore the same running water chan∣geth measure, when it changeth velocity; namely, increaseth the measure, when it decreaseth the velocity, and decreaseth the measure, when it increaseth the velocity.

On which principally depends all that which hath been said above in the Discourse, and observed in the Corollaries and Ap∣pendixes; and therefore is worthy to be well understood and heeded.

PROPOSITION IV.

If a River fall into another River, the height of the first in its own Chanel shall be to the height that it shall make in the second Chanel, in a proportion compounded of the proportions of the breadth of the Chanel of the second, to the breadth of the Chanel of the first, and of the velocitie acquired in the Chanel of the second, to that which it had in its proper and first Chanel.

LEt the River AB, whose height is AC, and breadth CB, that is, whose Section is ACB; let it enter, I say, into a∣nother River as broad as the line EF, and let it therein make the rise or height DE, that is to say, let it have its Section in the River whereinto it falls DEF; I say, that the height AC hath to the height DE the proportion compounded of the pro∣portions of the breadth EF, to the breadth CB, and of the ve∣locity through DF, to the velocity through AB. Let us sup∣pose the Section G, equal in velocity to the Section AB, and in breadth equal to EF, which carrieth a quantity of Water e∣qual to that which the Section AB carrieth, in equal times, and consequently, equal to that which DF carrieth. Moreover, as the breadth EF is to the breadth CB, so let the line H be to

the line I; and as the velocity of DF is to the velocity of AB, so let the line I be to the line L; because therefore the two Sections AB and G are equally swift, and discharge equal quan∣tity of Water in equal times, they shall be equal Sections; and

[illustration]

therefore the height of AB to the height of G, shall be as the breadth of G, to the breadth of AB, that is, as EF to CB, that is, as the line H to the line I: but because the Water which passeth through G, is equal to that which passeth through DEF, therefore the Section G, to the Section DEF, shall have the re∣ciprocal proportion of the velocity through DEF, to the velo∣city through G; but also the height of G, is to the height DE, as the Section G, to the Section DEF: Therefore the height of G, is to the height DE, as the velocity through DEF, is to the velocity through G; that is, as the velocity through DEF, is to the velocity through AB; That is, finally, as the line I, to the line L; Therefore, by equal proportion, the height of AB, that is, AC, shall be to the height DE; as H to L, that is, com∣pounded of the proportions of the breadth EF, to the breadth CB, and of the velocity through DF, to the velocity through A B: So that if a River fall into another River, &c. which was to be demonstrated.

PROPOSITION V.

If a River discharge a certain quantitie of Water in a certain time; and after that there come into it a Flood, the quantity of Water which is dischar∣ged in as much time at the Flood, is to that which was discharged before, whilst the River was low, in a proportion compounded of the proportions of the velocity of the Flood, to the velocity of the first Water, and of the height of the Flood, to the height of the first Water.

SUppose a River, which whilst it is low, runs by the Section AF; and after a Flood cometh into the same, and runneth through the Section DF, I say, that the quantity of the Wa∣ter which is discharged through DF, is to that which is discharged

[illustration]

through AF, in a proportion compounded of the proportions of the velocity through DF, to the velocity through AF, and of the height DB, to the height AB; As the velocity through DF is to the velocity through AF; so let the line R, to the line S; and as the height DB is to the height AB, so let the line S, to the line T; and let us suppose a Section LMN, equal to DF in height and breadth; that is LM equal to DB, and MN equal to BF; but let it be in velocity equal to the Section AF, there∣fore the quantity of Water which runneth through DF, shall be to that which runneth through LN, as the velocity through DF, is to the velocity through LN, that is, to the velocity through AF; and the line R being to the line S, as the velocity through DF, to the velocity through AF; therefore the quantity which runneth through DF, to that which runneth through LN, shall have the proportion of R to S; but the quantity which runneth through LN, to that which runneth through AF, (the Sections

being equally swift) shall be in proportion as the Section LN, to the Section AF; that is, as DB, to AB; that is as the line S, to the line T: Therefore by equal proportion, the quantity of the water which runneth through DF, shall be in proportion to that which runneth through AF, as R is to T; that is, compounded of the proportions of the height DB, to the height AB, and of the velocity through DF, to the velocity through AF; and therefore if a River discharge a certain quantity, &c. which was to be de∣monstrated.

ANNOTATION.

THe same might have been demonstrated by the second Proposition above demonstrated, as is manifest.

PROPOSITION VI.

If two equal streams of the same Torrent, fall into a River at divers times, the heights made in the Ri∣ver by the Torrent, shall have between them∣selves the reciprocal proportion of the velocities acquired in the River.

LEt A and B, be two equal streams of the same Torrent, which falling into a River at divers times, make the heights CD, and FG; that is the stream A, maketh the height CD, and the stream B, maketh the height FG; that is, Let their Sections in the River, into which they are fallen, be CE, and FH; I say, that the height CD, shall be to the height FG, in reciprocal proportion, as the velocity through FH, to the ve∣locity through CE; for the quantity of water which passeth through A, being equal to the quantity which passeth through B, in equal times; also the quantity which passeth through CE, shall

[illustration]

be equal to that which passeth through F H: And therefore the proportion that the Section CE, hath to the Section FH; shall be the same that the velocity through FH, hath to the velocity through CE; But the Section CE, is to the Section FH, as CD, to FG, by reason they are of the same breadth: Therefore CD, shall be to FG, in reciprocal proportion, as the velocity through FH, is to the velocity through CE, and therefore if two equal streams of the same Torrent, &c. which was to be de∣monstrated.