impediments removed the same water gaineth velocity and therefore decreaseth in measure and consequently in height And perhaps this point well understood may be of great profit to the fields adjacent to the Pontine Fens and I doubt not but if the River Ninfa and the other principal Brooks of those Territories were kept well cleansed from weeds their waters would be at a lower level and consequently the drains of the fields would run into them more readily it being alwayes to be held for undoubted that the measure of the water before the cleansing hath the same proportion to the measure after cleansing that the velocity after the cleansing hath to the velocity before the cleansing And because those weeds being cleansed away the course of the water notably increaseth it is therefore necessary that the said water abate in measure and become lower APPENDIX X. WE having above observed some errors that are committed in distributing the waters of Fountains and those that serve to water fields it seemeth now fit by way of a close to this discourse to advertise by what means these divisions may be made justly and without error I therefore think that one might two several wayes exquisitly divide the water of Fountains The first would be by diligently examining First how much water the whole Fountain dischargeth in a determinate time as for instance How many Barrels or Tuns it carrieth in a set time and in case you are afterwards to distribute the water distribute it at the rate of somany Barrels or Tuns in that same time and in this case the participants would have their punctual shares Nor could it ever happen to send out more water than is reckoned to be in the principal Fountain as befel Giulio Frontino and as also it frequently happeneth in the Modern Aqueducts to the publick and private detriment The other way of dividing the same waters of a Fountain is also sufficiently exact and easie and may be by having one onely size for the Cock or Pipe as suppose of an inch or of half an inch and when the case requireth to dispence two three and more inches take so many Cocks of the said measure as do evacuate the water which is to be emitted and if we are to make use onely of one greater Cock we being to place one to discharge for example four inches and having the former sole measure of an inch we must make a Cock that is bigger its true than the Cock of one inch but not simply in a quadruple proportion for that it would discharge more than just so much water as hath been said above but we ought to examine diligently how much water the little Cock emitteth in an hour and then enlarge and contract the greater Cock so that it may discharge four times as much water as the lesser in the same time and by this means we shall avoid the disorder hinted in the seventh Appendix It would be necessary nevertheless to accommodate the Cocks of the Cistern so that the level of the water in the Cistern may alwayes rest at one determinate mark above the Cock otherwise the Cocks will emit sometimes greater and sometimes lesse abundance of water And because it may be that the same water of the Fountain may be some times more abundant sometimes less in such case it will be necessary to adjust the Cistern so that the excess above the ordinary water discharge into the publick Fountains that so the particular participants may have alwayes the same abundance of water APPENDIX XI MUch more difficult is the division of the waters which serve to water the fields it not being possible to observe so commodiously what quantity of water the whole Ditch sends forth in one determinate time as may be done in Fountains Yet nevertheless if the second proposition by us a little below demonstrated be well understood there may be thence taken a very safe and just way to distribute such waters The Proposition therefore by us demonstrated is this If there be two Sections namely two mouths of Rivers the quantity of the water which passeth by the first hath a proportion to that which passeth by the second compounded of the proportions of the first Section to the second and of the velocity through the first to the velocity through the second As I will declare for example by help of practice that I may be understood by all in a matter so important Let the two mouths of the Rivers be A and B and let the mouth A be in measure and content thirty two feet and the mouth B eight feet Here you must take notice that it is not alwayes true that the Water which passeth by A hath the same proportion to that which passeth by B that the mouth A hath to the mouth B but onely when the velocityes by each of those passages are equal But if the velocityes shall be unequal it may be that the said mouths may emit equal quantity of Water in equal times though their measure be unequal and it may be also that the bigger doth discharge a greater quantity of Water And lastly it may be that the less mouth dischargeth more Water than the greater and all this is manifest by the things noted in the beginning of this discourse and by the said second Proposition Now to examine the proportion of the Water that passeth by one Ditch to that which passeth by another that this being known the same Waters and mouths of Ditches may be then adjusted we are to keep account not onely of the greatness of the mouths or passages of the Water but of the velocity also which we will do by first finding two numbers that have the same proportion between themselves as have the mouths which are the numbers 32 and 8 in our example Then this being done let the velocity of the Water by the passages A and B be examined which may be done keeping account what space a piece of Wood or other body that swimmeth is carried by the stream in one determinate time as for instance in 50 pulses and then work by the golden Rule as the velocity by A is to the velocity by B so is the number 8 to another number which is 4. It is clear by what is demonstrated in the said second Proposition that the quantity of water which passeth by the mouth A shall have the same proportion of that which passeth by the mouth B that 8 hath to 1. Such proportion being composed of the proportions of 32 to 8 and of 8 to 4 namely to the greatness of the mouth A to the greatness of the mouth B and of the velocity in A to the velocity in B. This being done we must then contract the mouth which dischargeth more then its just quantity of water or enlarge the other which dischargeth less as shal be most commodious in practice which to him that hath understood this little that hath been delivered will be very easie APPENDIX XII THese
never ceaseth to move away and as infinite is judged incomprehensible and such as that there is no exact knowledge to be had thereof so there comes to be no account made thereof but if we should make strict reflection upon our consideration of the velocity of Water we should find that keeping account of the same there is a reckoning also made of the length forasmuch as whilst we say the Water of such a Spring runs with the velocity of passing a thousand or two thousand paces an hour this in substance is no other than if we had said such a Fountain dischargeth in an hour a Water of a thousand or two thousand paces long So that albeit the total length of Running water be incomprehensible as being infinite yet neverthelesse it s rendered intelligible by parts in its velocity And so much sufficeth to have hinted about this matter hoping to impart on some other occasion other more accurate Observations in this affair LAVS DEO GEOMETRICAL DEMONSTRATIONS OF THE MEASURE OF Running Waters BY D. BENEDETTO CASTELLI Abbot of CASSINA and Mathematician to P. VRBAN VIII DEDICATED To the most Illustrious and most Excellent Prince DON THADDEO BARBERINI PRINCE OF PALESTRINA AND GENERAL of the HOLY CHURCH LONDON Printed Anno Domini MDCLXI 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 upper 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 quantity of the Water that passeth through the first Section shall have the same proportion to the quantity that passeth 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 velocity PROPOSITION I. The Sections of the same River discharge equal quantities of Water in equal times although the Sections 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 quantity of Water in equal times for if greater quantity of Water should pass through A than passeth through B it would 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 Section B is equal to the quantity of Water which passeth through the Section A and therefore the Sections of the same River discharge 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 passeth 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 proportions of the first Section A to the Section B and of the velocity 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 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 Section 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 proportion 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 line E to the line F therefore the quantity of water which passeth 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 velocity 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 velocity 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 velocities 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 changeth 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 Appendixes 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 another 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 proportions of the breadth EF to the breadth CB and of the velocity through DF to the velocity through AB Let us suppose the Section G equal in velocity to the Section AB and in breadth equal to EF which carrieth a quantity of Water equal 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 quantity of Water in equal times they shall be equal Sections and 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 reciprocal proportion of the velocity through DEF to the velocity 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 compounded 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 discharged 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 Water which is discharged through DF is to that which is discharged 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 therefore 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 demonstrated 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 River by the Torrent shall have between themselves 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 velocity 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 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 demonstrated OF THE MENSURATION OF Running Waters Lib. II. HAving in the close of my Treatise of the Mensuration of Running Waters promised to declare upon another occasion other particulars more obscure and of very great concern upon the same argumement I now do perform my promise on the occasion that I had the past year 1641. to propound my thoughts touching the state of the Lake of Venice a business certainly most important as being the concernment of that most noble and most admirable City and indeed of all Italy yea of all Europe Asia Africa one may truly say of all the whole World And being to proceed according to the method necessary in Sciences I wil propose in the first place certain Definitions of those Terms whereof we are to make use in our Discourse and then laying down certain Principles we will demonstrate some Problemes and Theoremes necessary for the understanding of those things which we are to deliver and moreover recounting sundry cases that have happened we will prove by practice of what utility this contemplation of the Measure of Running Waters is in the more important affairs both Publique and Private DEFINITION I. TWo Rivers are said to move with equal velocity when in equal times they passe spaces of equal length DEFINITION II. RIvers are said to move with like velocity when their proportional parts do move alike that is the upper parts alike to the upper and the lower to the lower so that if the upper part of one River shall be more swift than the upper part of another then also the lower part of the former shall be more swift than the part correspondent to it in the second proportionally DEFINITION III. TO measure a River or running Water is in our sense to finde out how many determinate measures or weights of Water in a given time passeth through the River or Channel of the Water that is to be measured DEFINITION IV. IF a Machine be made either of Brick or of Stone or of Wood so composed that two sides of the said Machine be placed at right angels upon the ends of a third side that is supposed to be placed in the bottom of a River parallel to the Horizon in such a manner that all the water which runneth through the said River passeth thorow the said Machine And if all the water coming to be diverted that runneth through the said River the upper superficies of that third side placed in the bottom do remain uncovered and dry and that the dead water be not above it This same Machine shall be called by us REGULATOR And that third side of the Machine which standeth Horizontally is called the bottom of the Regulator and the other two sides are called the banks of the Regulator as is seen in this first Figure ABCD shall be the Regulator BC the bottom and the other two sides AB and CD are its banks DEFINITION V. BY the quick height we mean the Perpendicular from the upper superficies of the River unto the upper superficies of the bottom of the Regulator as in the foregoing Figure the line GH DEFINITION VI. IF the water of a River be supposed to be marked by three sides of a Regulator that Rightangled Parallelogram comprehended between the banks of the Regulator and the bottom and the superficies of the Water is called a Section of the River ANNOTATION HEre it is to be noted that the River it self may have sundry and divers heights in several parts of its Chanel by reason of the various velocities of the water and its measures as hath been demonstrated in the first book SUPPOSITION I. IT is supposed that the Rivers equal in breadth and quick height that have the same inclination of bed or bottom ought also to have equal velocities the accidental impediments being removed that are dispersed throughout the course of the water and abstracting also from the external windes which may velocitate and retard the course of the water of the River SUPPOSITION II. LEt us suppose also that if there be two Rivers that are in their beds of equal length and of the same inclination but of quick heights unequal they ought to move with like velocity according to the
the whole greater square as the water of the Chanellet diverted hath to the water of the bigger Chanel And because the water of the Chanellet is known by the Method laid down in the first Theorem and the terms of the Theorem being also known the quantity of the water which runneth thorow the bigger Chanel shall be also known by the Golden Rule which was that that was desired to be known We will explain the whole business by an example Let a Chanel be for example 15 Palms broad its quick height before its diversion by the Chanellet shall be supposed to be 24 inches but after the diversion let the quick height of the Chanel be onely 22 inches Therefore the greater height to the lesser is as the number 11. to 12. But the square of 11. is 121 and the square of 12. is 144 the difference between the said lesser square and the greater is 23. Therefore the diverted water is to the whole water as 23. to 144 which is well near as 1 to 6 6 23 and that is the proportion that the quantity of the water which runneth through the Chanellet shall have to all the water that runneth thorow the great Chanel Now if we should finde by the Rule mentioned above in the first Proposition that the quantity of the water that runneth through the Chanellet is v. g. an hundred Barrels in the space of 15 second minutes of an hour it is manifest that the water which runneth through the great Chanel in the said time of 35 min. sec shall be about 600 Barrels The same operation performed another way ANd because very often in applying the Theory to Practice it happeneth that all the necessary particulars in the Theory cannot so easily be put in execution therefore we will here add another way of performing the same Problem if it should chance to happen that the Chanellet could not commodiously be diverted from the great Chanel but that it were easier for the water of another smaller Chanel to be brought into the greater Chanel which water of the smaller Chanel might be easily measured as hath been shewen in the first Probleme or in case that there did fall into a greater Chanel a lesser Chanel that might be diverted and measured Therefore I say in the first case If we would measure the quantity of the water that runneth in a certain time thorow the greater Chanel into which another lesser Chanel that is measurable may be brought we must first exactly measure the Chanellet and then observe the quick height of the greater Chanel before the introduction of the lesser and having brought in the said Chanellet we must agnin find the proportion that the water of the Chanellet hath to all the water of the great Chanel for these terms of the proportion being known as also the quantity of the water of the Chanellet we shall also come to know the quantity of the water that runneth thorow the great Chanel It is likewise manifest that we shall obtain our intent if the case were that there entered into the great Chanel another lesser Chanel that was measurable and that might be diverted CONSIDERATION IT would be necessary to make use of this Doctrine in the distribution of the waters that are imploy'd to overflow the fields as is used in the Brescian Cremonese Bergamase Lodigian Milanese territories and many other places where very great suits and differences arise which not being to be determined with intelligible reasons come oftentimes to be decided by force of armes and instead of flowing their Grounds with Waters they cruelly flow them with the shedding of humane blood impiously inverting the course of Peace and Justice sowing such disorders and feuds as that they are sometimes accompanied with the ruine of whole Cities or else unprofitably charge them with vain and sometimes prejudicial expences PROPOS IV. THEOR. II. If a River increase in quick height the quantitie of Water which the River dischargeth after the increase hath the Proportion compounded of the Proportions of the Quick height to the Quick height and of the velocity to the velocity LEt there be a River which whilst it is low runneth thorow the Regulator DF with the Quick height AB and afterwards let a Flood come and then let it run with the height DB I say that the quantity of the Water that is discharged through DF to that which dischargeth through AF hath the 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 be to the line S and as the height DB is to the height AB so let the line S be to the line T. And let a Section be supposed LMN equal to the Section DF in height and length but let it be in velocity equal to the Section AF. Therefore the quantity of the Water that runneth through DF to that which runneth through LN shall be as the velocity through DF to the velocity of LN that is to the velocity through LN that is to the velocity through AF. therefore the quantity of Water which runneth through DF to that which passeth through LN shall have the proportion that R hath to S but the quantity of the Water that runneth through LN to that which runneth through AF the Sections being equally swift shall have the proportion that the Section LN hath to the Section AF that is that the height BD hath to the height BA that is that S hath to T. Therefore by equal proportion the quantity of the Water which runneth by DF to that which runneth by AF shall have the proportion of R to T that is shall be 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 increase in quick height the quantity of the Water that runneth after the increase to that which runneth before the increase hath the proportion compounded c. Which was to be demonstrated COROLLARIE I. HEnce it followeth that we having shewn that the quantity of the Water which runneth whilst the River is high to that which ran whilst it was low hath the proportion compounded of the velocity to the velocity and of the height to the height And it having been demonstrated that the velocity to the velocity is as the height to the height it followeth I say that the quantity of the Water that runneth whilst the River is high to that which runneth whilst it is low hath duplicate proportion of the height to the height that is the proportion that the squares of the heights have COROLLARIE II. VPon which things dependeth the reason of that which I have said in my second Consideration that if by the diversion of 5 9 of the Water that entereth by the Rivers into the Moor or Fen the Water
subject of Philosophy 1 Mud. Vide Sand. N Navigation from Bologna to Ferrara is become impossible till such time as Reno be diverted 101 Navigation in the Lake of Venice endangered and how restored 65 70 P Perpendicularity of the Banks of the River to the upper superficies of it 37 Perpendicularity of the Banks to the bottom 37 Perugia Vide Lake Pontine Vide Fenns Ports of Venice Malamocco Bondolo and Chiozza choaked up for want of Water in the Lake 65 Proportions of unequal Sections of equal Velocity and of equal Sections of unequal Velocity Axiome IV. and V. 38 Proportions of equal and unequal quantities of Water which pass by the Sections of different Rivers Proposition II. 39 Proportions of unequal Sections that in equal times discharge equal quantities of Water Proposition III. 41 Proportion wherewith one River falling into another varieth in height Proposition IV. 44 Proportion of the Water discharged by a River in the time of Flood to the Water discharged in an equal time by the said River before or after the Flood Proposition V. 44 Proportion of the Heights made by two equal Brooks or Streams falling into the same River Proposition VI. 45 Proportion of the Water which a River dischargeth encreasing in Quick-height by the addition of new Water to that which it dischargeth after the encrease is made Proposition IV. Theor. II. 54 Proportion of a River when high to it self when low Coroll I. 55 Q Quantity of Running Waters is never certain if with the Vulgar way of Measuring them their Velocities be not considered 32 Quantities of Waters which are discharged by a River answer in equality to the Velocities and times in which they are discharged Axiome I II III. 38 Quick-Height of a River what it is Definition V. 48 R Reason of the Proverb Take heed of the still Waters Coroll VI. 7 Reasons of Monsignore Corsini against the diversion of Reno into the Po of Volano 105 Reasons of Cardinal Capponi and Monsig Corsini for the turning of Reno into Main Po. 106 Two objections on the contrary and answers to them 104 105 What ought to be the proportion of the Heights of Reno is Reno and of Reno in Po. 110 Regulator what it is Definition IV. 48 Relation of the Waters of Bologna and Ferrara by Monsignore Corsini 100 Reno in the Valleys and its bad effects 100 101 Two wayes to divert it 103 The facility and utility of those wayes Ibid. The difficulties objected 104 Reply to Bartolotti touching the dangers of turning Fiume Morto into Serchio 83 Retardment of the course of a River caused by its Banks Appendix VII 19 Risings made by Flood-Gates but small Appendix XIII 26 Rivers that are shallow swell much upon small showers such as are deep rise but little upon great Floods Corollary III. 6 Rivers the higher they are the swifter Ibid. Rivers the higher they are thelesse they encrease upon Floods 49 Rivers when they are to have equal and when like Velocity Ibid. Rivers in falling into the Sea form a Shelf of Sand called Cavallo 65 Five Rivers to be diverted from the Lake of Venice and the inconveniences that would ensue thereupon 74 75 A River of Quick-height and Velocity in its Regulator being given if the Height be redoubled by new Water it redoubleth also in Velocity Proposition II. Theorem I. 51 Keepeth the proportion of the heights to the Velocities Corollary 52 S Sand and Mud that entereth into the Lake of Venice and the way to examine it 76 Seas agitated and driven by the Winds stop up the Ports 64 65 Sections of a River what they are Definition I. 37 Sections equally swift what they are Definition II. Ibid. Sections of River being given to conceive others equal to them of different breadth height and Velocity Petition 38 Sections of the same River and their Proportions to their Velocities Coroll I. 42 Sections of a River discharge in any whatsoever place of the said River equal quantities of Water in equal times Proposition I. 39 Sile River what mischiefes it threatneth diverted from the Lake 74 Spirtings of Waters grow bigger the higher they go Coroll XVI 16 Streams of Rivers how they encrease and vary Coroll I. 6 Streams retarded and the effects thereof Coroll IX 8 T Table of the Heights Additions and Quantities of Waters and its use 56 Thrasimenus Vide Lake Time how it s measured in these Operations of the Waters 49 Torrents encrease at the encreasing of a River though they carry no more Water than before Coroll IV. 6 Torrents when they depose and carry away the Sand. Coroll V. 7 Torrents and their effects in a River 6 7 Torrents that fall into the Valleys or into Po of Volano and their mischiefs prevented by the diverting of Reno into Main Po. 100 Tyber and the causes of its inundations Coroll VIII 8 V Valleys of Bologna and Ferrara their inundations and disorders whence they proceed 97 Velocity of the Water shewn by several Examples 3 Its proportion to the Measure 5 Velocities equal what they are 47 Velocities like what they are 47 48 Velocities of Water known how they help us in finding the Lengths 113 A Fable to explain the truth thereof Ibid. Venice Vide Lake Vse of the Regulator in measuring great Rivers Consideration I. 60 W Waters falling why they disgross Coroll XVI 16 Waters how the Length of them is Measured 70 Waters that are imployed to flow Grounds how they are to be distributed 19 53 54 Waters to be carryed in Pipes to serve Aquaducts and Conduits how they are to be Measured 115 116 Way to know the rising of Lakes by Raines 28 Way of the Vulgar to Measure the VVaters of Rivers 68 Wind Gun and Portable Fountain of Vincenzo Vincenti of Urbin 11 Windes contrary retard and make Rivers encrease Coroll VII 8 The END of the TABLE of the Second Part of the First TOME a Nella continuatione dell Nuntio â—iderio b Lettera al P. Abbate D. B. Castelli D'Arcetro li. 3. Decemb. 16â—9 c De Motu Aquarum Lib. 2. Prop. 37. p. 191. * * And as is at large demonstrated by that most excellent and Honourable personage Mr. Boile in the industrious experiment of his Pneumatical Engine * * Artesia * * Commentarius beareth many senses but in this place signifieth a certain Register of the quantities of the Waters in the several publique Aquiducts of Rome which word I find frequently used in the Law-books of antient Civilians And by errogation we are to understand the distribution or delivering out of those stores of Water * * A Coyn of Pope Julius worth six pence * * Or Sluice * * In Pregadi a particular Council the Senators of which have great Authority * * A Venice Brace is 11 1â— of our yard * * A River of that name * * I. Savii dell ' Acque a particular Council that take care of the Lakes and other Aquatick affairs * * He here intends the Demonstrations following at the end of the first Book * * Deeper * * Lib. 1. * * The Countrey or Province lying round the City heretofore called Lati 〈◊〉 * * Or Lordship * * The Popes Exchequer * * Polesine is a plat of Ground almost surrounded with Bogs or waters like an Island * * People of Ferrara * * In Chanels made by hand * * The inch of these places is somewhat bigger than ours * * Of Adriano * Larghezza but misprinted
profitable a matter and to supply also my defects in this short and difficult Tractate Difficult I say for the truth is these knowledges though of things next our senses are sometimes more abstruce and hidden then the knowledge of things more remote and much better and with greater exquisitness are known the motions of the Planets and Periods of the Stars than those of Rivers and Seas As that singular light of Philosophie of our times and my Master Signore Galileo Galilei wisely observeth in his Book concerning the Solar spots And to proceed with a due order in Sciences I will take some suppositions and cognitions sufficiently clear from which I will afterwards proceed to the deducing of the principal conclusions But to the end that what I have written at the end of this discourse in a demonstrative and Geometrical method may also be understood of those which never have applyed their thoughts to the study of Geometry I have endeavoured to explain my conceit by an example and with the consideration of the natural things themselves must after the same order in which I began to doubt in this matter and have placed this particular Treatise here in the beginning adverting nevertheless that he who desires more full and absolute solidity of Reasons may overpass this prefatory discourse and onely consider what is treated of in the demonstrations placed towards the end and return afterwards to the consideration of the things collected in the Corollaries and Appendices which demonstrations notwithstanding may be pretermitted by him that hath not seen at least the first six Books of the Elements of Euclid so that he diligently observeth that which followeth I say therefore that having in times past on divers occasions heard speak of the measures of the waters of Rivers and Fountains saying such a River is two or three thousand feet of water such a spring-water is twenty thirty or forty inches c. Although in such manner I have found all to treat thereof in word and writing without variety and as we are wont to say constanti sermone yea even Artists and Ingeneers as if it were a thing that admitted not of any doubt yet howsoever I remained still infolded in such an obscurity that I well knew I understood nothing at all of that which others pretended full and clearly to understand And my doubt arose from my frequent observation of many Trenches and Channels which carry water to turn Mills in which Trenches and Channels the water being measured was found pretty deep but if afterwards the same water was measured in the fall it made to turn the Wheel of the Mill it was much lesse not amounting often to the tenth part nor sometimes to the twentieth insomuch that the same running water came to be one while more another while less in measure in divers parts of its Channel and for that reason this vulgar manner of measuring running Waters as indeterminate and uncertain was by me justly suspected the measure being to be determinate and the same And here I freely confesse that I had singular help to resolve this difficulty from the excellent accurate way of discoursing as in allother matters so also in this of the Right Honourable and Truly Noble Signior Ciampoli Secretary of the Popes secret affairs Who moreover not sparing for the costs of the same generously gave me occasion a few years past to try by exact experiments that which past concerning this particular And to explain all more clearly with an example we suppose a Vessel filled with Water as for instance a Butt which is kept full though still water runneth out and the Water run out by two Taps equal of bignesse one put in the bottom of the Vessel and the other in the upper part it is manifest that in the time wherein from the upper part shall issue a determinate measure of water from the inferiour part there shall issue four five and many more of the same measures according to the difference of the height of the Taps and the distance of the upper Tap from the Superficies and level of the water of the Vessel and all this will alwayes follow though as hath been said the Taps be equal and the water in discharging keep the said Taps alwayes full Where first we note that although the measure of the Taps be equal neverthelesse there issueth from them in equal times unequal quantities of water And if we should more attentively consider this businesse we should find that the water by the lower Tap runneth and passeth with much greater velocity then it doth by the upper whatever is the reason If therefore we would have such a quantity of Water discharge from the upper tap as would discharge from the neather in the same time it is plain that either the upper Tapâ— must be multiplyed in such sort that so many more Taps in number be placed above than below as the neather tap shall be more swift than the upper or the upper Tap made so much bigger than the nether by how much that beneath shall be more swift than that above and so then in equal times the same quantity of Water shall discharge from the upper as doth from the neather part I will declare my self by another example If we should imagine that two cords or lines of equal thickness be drawn through two holes of equal bore but so that the first pass with quadruple velocity to the second It is manifest that if in a determinate time we shall by the first bore have drawn four Ells of the line in the same time by the second hole we shall have drawn but one Ell of cord onely and if by the first there passe twelve Ells then through the second there shall passe onely three Ells and in short the quantity of cord shall have the same proportion to the cord that the volocity hath to the velocity And therefore we desiring to compensate the tardity of the second cord and maintaining the same tardity to draw through the second hole as much cord as through the first it will be necessary to draw through the second bore four ends of cord so that the thickness of all the cords by the second hole have the same proportion to the thickness of the cord which passeth onely by the first as the velocity of the cord by the first hole hath reciprocally to the velocity of the cod's by the second hole And thus its clear that when there is drawn through two holes equal quantity of cords in equal time but with unequal velocity it will be necessary that the thickness of all the four cords shall have the same reciprocal proportion to the thickness of the swifter cord that the velocity of the swifter cord hath to the velocity of the slower The which is verified likewise in the fluid Element of Water And to the end that this principal fundamental be well understood I will also note a certain observation made my me in the Art of
Wyer-drawing or spinning Gold Silver Brass and Iron and it is this That such Artificers desiring more and more to disgrosse and subtillize the said Metals having wound about a Rocket or Barrel the thread of the Metal they place the Rocket in a frame upon a stedfast Axis in such sort that the Rocket may turn about in it self then making one end of the thread to passe by force through a Plate of Steel pierced with divers holes greater and lesser as need requireth fastning the same end of the thread to another Rocket they wind up the thread which passing through a bore less than the thicknesse of the thread is of force constrained to disgrosse and subtillize Now that which is intensly to be observed in this business is this That the parts of the thread before the hole are of such a thicknesse but the parts of the same thread after it is passed the hole are of a lesser thicknesse and yet neverthelesse the masse and weight of the thread which is drawn forth is ever equal to the masse and weight of the thread which is winded up But if we should well consider the matter we should finde that the thicker the thread before the hole is than the thread passed the hole the greater reciprocally is the velocity of the parts of the thread passed the hole than the volocity of the parts before the hole Insomuch that if vâ—rbi gratia the thicknesse of the thread before the hole were double to the thicknesse after the hole in such case the velocity of the parts of the thread passed the hole should be double to the velocity of the parts of the thread before the hole and thus the thicknesse compensates the velocity and the velocity compensates the thicknesse So that the same occurreth in the solid Metals of Gold Silver Brass Iron c. that eveneth also in the fluid Element of Water and other liquids namely That the velocity beareth the same proportion to the velocity that the thicknesse of the Metal or Water hath to the thicknesse And therefore granting this discourse we may say that as often as two Taps with different velocity discharge equal quantities of Water in equal times it will be necessary that the Tap lesse swift be so much greater and larger than the Tap more swift by how much the swifter superates in velocity the slower and to pronounce the Proposition in more proper terms we say That if two Taps of unequal velocity discharge in equal times equal quantities of Water the greatnesse of the first shall be to the greatnesse of the second in reciprocal proportion as the velocity of the second to the velocity of the first As for example if the first Tap shall be ten times swifter than the second Tap it will be necessary that the second be ten times bigger and larger than the first and in such case the Taps shall discharge equall quantities of water in equal times and this is the principal and most important point which ought to be kept alwayes in minde for that on it well understood depend many things profitable and worthy of our knowledge Now applying all that hath been said neerer to our purpose I consider that it being most true that in divers parts of the same River or Current of running water there doth always passe equal quantity of Water in equal time which thing is also demonstrated in our first Proposition and it being also true that in divers parts the same River may have various and different velocity it follows of necessary consequence that where the River hath lesse velocity it shall be of greater measure and in those parts in which it hath greater velocity it shall be of lesse measure and in sum the velocity of several parts of the said River shall have eternally reciprocall and like proportion with their measures This principle and fundamental well established that the same Current of Water changeth measure according to its varying of velocity that is lessening the measure when the velocity encreaseth and encreasing the measure when the velocity decreaseth I passe to the consideration of many particular accidents in this admirable matter and all depending on this sole Proposition the sense of which I have oft repeated that it might be well understood COROLLARIE I. ANd first we hence conclude that the same Streams of a Torrent namely those streams which carry equal quantity of Water in equal times make not the same depths or measures in the River in which they enter unlesse when in the entrance into the River they acquired or to say better keep the same velocity because if the velocicities acquired in the River shall be different also the measures shall be diverse and consequently the depths as is demonstrated COROLLARIE II. ANd because successively as the River is more and more full it is constituted ordinarily in greater greater velocity hence it is that the same streams of the Torrent that enter into the River make lesse and lesse depths as the River grows more and more full since that also the Waters of the Torrent being entered into the River go acquiring greater and greater velocities and therefore diminish in measure and height COROLLARIE III. WE observe also that while the main River is shallow if there fall but a gentle rain it suddenly much increaseth and riseth but when the River is already swelled though there fall again another new violent shower yet it increaseth not at the same rate as before proportionably to the rain which fell which thing we may affirm particularly to depend on this that in the first case while the River is low it is found also very slow and therefore the little water which entereth into it passeth and runs with little velocity and consequently occupieth a great measure But when the River is once augmented by new water being also made more swift it causeth the great Flood of water which falleth to bear a lesse measure and not to make such a depth COROLLARIE IV. FRom the things demonstrated is manifest also that whilst a Torrent entereth into a River at the time of Ebbe then the Torrent moveth with such a certain velocity what ever it be passing by its extreamest parts wherewith it communicateth with the River in which parts the Torrent being measured shall have such a certain measure but the River swelling and rising also those parts of the Torrent augment in greatnesse and measure though the Torrent in that instant dis-imbogue no more water than it did before so that the River being swelled we are to consider two mouths of the same Torrent one lesse before the rising the other greater after the rising which mouths discharge equal quantities of water in equal times therefore the velocity by the lesser mouth shall be greater than the velocity by the greater mouth and thus the Torrent shall be retarded from its ordinary course COROLLARIE V. FRom which operation of Nature proceedeth another effect worthy of consideration and it
Body is known and the quantity of a Body being given if you have but two dimensions the third shall be known And thus diving farther and farther into this Consideration I found that the Velocity of the course of the water may be an hundred times greater or lesser in one part of its Chanel than in another And therefore although there should be two mouths of Waters equal in bignesse yet nevertheless it might come to passe that one might discharge an hundred or a thousand-times more water than another and this would be if the water in one of the mouths should run with an hundred or a thousand times greater velocity than the other for that it would be the same as to say that the swifter was an hundred or a thousand times longer than the slower and in this manner I discovered that to keep account of the velocity was the keeping account of the Length And therefore it is manifest that when two Mouths discharge the same quantity of Water in an equal velocity it is necessary that the less swift Mouth be so much bigger than the more swift as the more swift exceedeth in velocity the less swift as for example In case two Rivers should carry equal quantity of water in equal times but that one of them should be four times more swift than the other the more slow should of necessity be four times more large And because the same River in any part thereof alwaies dischargeth the same quantity of Water in equal times as is demonstrated in the first Proposition of the first Book of the measure of Running Waters but yet doth not run thorowout with the same velocity Hence it is that the vulgar measures of the said River in divers parts of its Chanel are alwaies divers insomuch that if a River passing through its chanel had such velocity that it ran 100 Braces in the 1 60 of an hour-and afterwards the said River should be reduced to so much tardi ty of motion as that in the same time it should not run more than one Brace it would be necessary that that same River should become 100. times bigger in that place where it was retarded I mean 100. times bigger than it was in the place where it was swifter And let it be kept well in mind that this point rightly understood will clear the understanding to discover very many accidents worthy to be known But for this time let it suffice that we have onely declared that which makes for our purpose referring apprehensive and studious Wits to the perusal of my aforenamed Treatise for therein he shall finde profit and delight both together Now applying all to our principal intent I say That by what hath been declared it is manifest that if the Brent were 48. Braces broad and 2 ½ high in some one part of its Chanel that afterwards the same Water of the Brent falling into the Lake and passing thorow the same to the Sea it should lose so much of its velocity that it should run but one Brace in the time wherein whilst it was in its Chanel at the place aforesaid it ran 100. Braces It would be absolutely necessary that increasing in measure it should become an hundred times thicker and therefore if we should suppose that the Lake were 20000. Braces the Brent that already hath been supposed in its Chanel 100. Braces being brought into the Lake should be 100. times 100. Brates that is shall be 10000. Braces in thickness and consequently shall be in height half a Brace that is 100 200 of a Brace and not 1 200 of a Brace as was concluded in the Argument Now one may see into what a gross errour of 99. in 100. one may fall through the not well understanding the true quantity of Running Water which being well understood doth open a direct way to our judging aright in this most considerable affair And therefore admitting that wich hath been demonstrated I say that I would if it did concern me greatly encline to consult upon the returning of the Brent again into the Lake For it being most evident that the Brent in the Chanel of its mouth is much swifter than the Brent being brought into the Lake it will certainly follow thereupon that the thickness of the Water of Brent in the Lake shall be so much greater than that of Brent in Brent by how much the Bront in Brent is swifter than thh Brent in the Lake 1. From which operation doth follow in the first place that the Lake being filled and increased by these Waters shall be more Navigable and passible than at present we see it to be 2. By the current of these Waters the Chanels will be scoured and will be kept clean from time to time 3. There will not appear at the times of low-waters so many Shelves and such heaps of Mud as do now appear 4. The Ayr will become more wholesom for that it shall not be so infected by putrid vapours exhaled by the Sun so long as the Miery Ouze shall be covered by the Waters 5. Lastly in the current of these advantagious Waters which must issue out of the Lake into the Sea besides those of the Tyde the Ports will be kept scoured and clear And this is as much as I shall offer for the present touching this weighty business alwaies submitting my self to sounder judgements Of the above-said Writing I presented a Copy at Venice at a full Colledge in which I read it all and it was hearkned to with very great attention and at last I presented it to the Duke and left some Copies thereof with sundry Senators and went my way promising with all intenseness to apply my pains with reiterated studies in the publick service and if any other things should come into my minde I promised to declare them sincerely and so took leave of His serenity and that Noble Council When I was returned to Rome this business night and day continually running in my mind I hapned to think of another admirable and most important conceit which with effectual reasons confirmed by exact operations I with the Divine assistance made clear and manifest and though the thing at first sight seemed to me a most extravagant Paradox yet notwithstanding having satisfied my self of the whole business I sent it in writing to the most Illustrious and most Noble Signore Gio. Basadonna who after he had well considered my Paper carried it to the Council and after that those Lords had for many months maturely considered thereon they in the end resolved to suspend the execution of the diversion which they had before consulted to make of the River Sile and of four other Rivers which also fall into the Lake a thing by me blamed in this second Paper as most prejudicial and harmful The writing spake as followeth CONSIDERATIONS Concerning the LAKE OF VENICE PART II. IF the discoursing well about the truth of things Most Serene Prince were as the carrying of Burdens
is that the course of the water retarding as hath been said in those ultimate parts of the Torrent if it shall happen that the Torrent grow torbid and muddy and its streame be retarded in such a degree that it is not able to carry away those minute grains of Earth which compose the muddinesse in this case the Torrent shall clear away the mud and carry away the Sand at the bottome of its own Chanel in the extream parts of its mouth which raised and voided Sand shall again afterwards be carried away when the River abating the Torrent shall return to move with its primitive velocity COROLLARIE VI. WHilst it is demonstrated that the same water hath different measures in its Chanel or course according as it varieth in velocity so that the measure of the water is alwayes greater where the velocity is lesser and on the contrary the measure lesser where the velocity is greater from hence we may most elegantly render the reason of the usual Proverb Take heed of the still waters For that if we consider the self same water of a River in those parts wherein it is less swift and thence called still or smooth water it shall be of necessity of greater measure than in those parts in which it is more swift and therefore ordinarily shall be also more deep and dangerous for passengers whence it is well said Take heed of the still Waters and this saying hath been since applied to things moral COROLLARIE VII LIkewise from the things demonstrated may be concluded that the windes which stop a River and blowing against the Current retard its course and ordinary velocity shall necessarily amplifie the measure of the same River and consequently shall be in great part causes or we may say potent con-causes of making the extraordinary inundations which Rivers use to make And it s most certain that as often as a strong and continual wind shall blow against the Current of a River and shall reduce the water of the River to such tardity of motion that in the time wherein before it run five miles it now moveth but one such a River will increase to five times the measure though there should not be added any other quantity of water which thing indeed hath in it something of strange but it is most certain for that look what proportion the waters velocity before the winde hath to the velocity after the winde and such reciprocally is the measure of the same water after the winde to the measure before the winde and because it hath been supposed in our case that the velocity is diminished to a fifth part therefore the measure shall be increased five times more than that which it was before COROLLARIE VIII WE have also probable the cause of the inundations of Tyber which befel at Rome in the time of Alexander the Sixth of Clement the Seventh which innundations came in a serene time and without great thaws of the Snows which therefore much puzzled the wits of those times But we may with much probability affirm That the River rose to such a height and excresence by the retardation of the Waters dependant on the boistrous and constant Winds that blew in those times as is noted in the memorials COROLLARIE IX IT being most manifest that by the great abundance of Water the Torrents may increase and of themselves alone exorbitantly swell the River and having demonstrated that also without new Water but onely by the notable retardment the River riseth and increaseth in measure in proportion as the velocity decreaseth hence it is apparent that each of these causes being able of it self and separate from the other to swell the River when it shall happen that both these two causes conspire the augmentation of the River in such a case there must follow very great and irrepable innundations COROLLARIE X. FRom what hath been demonstrated we may with facility resolve the doubt which hath troubled and still poseth the most diligent but incautelous observers of Rivers who measuring the Streams and Torrents which fall into another River as those for instance which enter into the Po or those which fall into Tiber and having summed the total of these measures and conferring the measures of the Rivers and Brooks which fall into Tiber with the measure of Tiber and the measures of those which disimbogue into Po with the measure of Po they find them not equal as it seems to them they ought to be and this is because they have not well noted the most important point of the variation of velocity and how that it is the most potent cause of wonderfully altering the measures of running Waters but we most facilly resolving the doubt may say that these Waters diminish the measure being once entered the principal Channel because they increase in velocity COROLLARIE XI THrough the ignorance of the force of the velocity of the Water in altering its measure augmenting it when the velocity diminisheth and diminishing it when the velocity augmenteth The Architect Giovanni Fontana endeavoured to measure and and to cause to be measured by his Nephew all the Brooks and Rivers which discharged their Waters into Tiber at the time of the Innundation which happened at Rome in the year 1598 and published a small Treatise thereof wherein he summeth up the measures of the extraordinary Water which fell into Tiber and made account that it was about five hundred Ells more than ordinary and in the end of that Treatise concludeth that to remove the Innundation wholly from Rome it would be necessary to make two other Channels equal to that at present and that lesse would not suffice and finding afterwards that the whole Stream passed under the Bridge Quattro-Capi the Arch whereof is of a far less measure then five hundred Ells concludeth that under the said Bridge past a hundred fifty one Ells of Water compressed I have set down the precise term of comprest Water written by Fontana wherein I finde many errors The first of which is to think that the measures of these Waters compressed in the Channels of those Brooks and Rivers should maintain themselves the same in Tiber which by his leave is most false when ever those waters reduced into Tiber retain not the same velocity which they had in the place in which Fontana and his Nephew measured them And all this is manifest from the things which we have above explained for if the Waters reduced into Tiber increase in velocity they decrease in measure and if they decrease in velocity they increase in measure Secondly I consider that the measures of those Brooks and Rivers which enter into Tiber at the time of Innundation are not between themselves really the same when their velocities are not equal though they have the same names of Ells and Feet for that its possible that a disinboguement of ten Ells requadrated to speak in the phrase of Fontana of one of those Brooks might discharge into Tiber at
the time of Innundation four ten and twenty times less Water than another mouth equal to the first in greatness as would occur when the first mouth were four ten or twenty times less swift than the second Whereupon whilst Fontana summes up the Ells and Feet of the measures of those Brooks and Rivers into a total aggregate he commits the same error with him which would add into one summe diverse moneys of diverse values and diverse places but that had the same name as if one should say ten Crowns of Roman money four Crowns of Gold thirteen Crowns of Florence five Crowns of Venice and eight Crowns of Mantua should make the same summe with forty Crowns of Gold or forty Crowns of Mantua Thirdly It might happen that some River or Current in the parts nearer Rome in the time of its flowing did not send forth more Water than ordinary and however it s a thing very clear that whilst the stream came from the superior parts that same Brook or River would be augmented in measure as hath been noted in the fourth Corollary in such sort that Fontana might have inculcated and noted that same River or Current as concurring to the Innundation although it were therein altogether unconcerned Moreover in the fourth place we must note That it might so fall out that such a River not onely was uninteressed in the Innundation though augmented in measure but it might I say happen that it was instrumental to the asswaging the Innundation by augmenting in the measure of its own Channel which matter is sufficiently evident for if it be supposed that the River in the time of flood had not had of it self and from its proper springs more Water than ordinary it s a thing certain that the Water of Tiber rising and increasing also that River to level it self with the Water of Tiber would have retained some of its Waters in its own Chanel without discharging them into Tyber or else would have ingorged and swallowed if I may so say some of the water of Tyber and in this case at the time of Inundation lesse abundance of water would have come to Rome and yet neverthelesse the measure of that River would have been increased Fifthly Fontana deceiveth himself when he concludeth that to remove the Inundation from Rome it would be necessary to make two other Chanels of Rivers that were as large as that which is the present one and that less would not suffice which I say is a fallacy and to convince him easily of his errour it sufficeth to say that all the Streams being passed under the Bridge Quattro-Capi as he himself attests a Channel would suffice only of the capacity of the said Bridge provided that the water there might run with the same velocity as it did under the Bridge at the time of Inundation and on the contrary twenty Currents of capacity equal to the present one would not suffice if the water should run with twenty times less velocity than it made at the time of the Inundation Sixthly to me it seemeth a great weaknesse to say that there should passe under the Bridge Quattro-Capi an hundred fifty one ells of water compressed for that I do not understand that water is like Cotton or Wool which matters may be prest and trod as it happeneth also to the air which receiveth compression in such sort that after that in some certain place a quantity of air shall be reduced to its natural constitution and having taken up all the said place yet neverthelesse compressing the first Air with force and violence it is reduced into far less room and will admit four or six times as much air as before as is experimentally seen in the Wind-Gun invented in our dayes by M. Vincenzo Vincenti of Vrbin which property of the Air of admitting condensation is also seen in the portable Fountains of the same M. Vincenzo which Fountains spirt the Water on high by force of the Air compressed which whilst it seeks to reduce its self to its natural constitution in the dilation causeth that violence But the water can never for any thing I know crowd or press so as that if before the compression it held or possest a place being in its natural constitution I believe not I say that it is possible by pressing and crowding to make it possess less room for if it were possible to compress the Water and make it to occupy a less place it would thence follow that two Vessels of equal measure but of unequal height should be of unequal capacity and that should hold more water which was higher also a Cylinder or other Vessel more high than broad would containe more water erected than being laid along for that being erected the water put therein would be more pressed and crowded And therefore in our case according to our principles we will say that the water of that Stream passeth all under the said Bridge Quattro-Capi for that being there most swift it ought of consequence to be less in measure And here one may see into how many errours a man may run through ignorance of a true and real Principle which once known and well understood takes away all mists of doubting and easily resolveth all difficulties COROLLARIE XII THrough the same inadvertency of not regarding the variation of velocity in the same Current there are committed by Ingineers and Learned men errours of very great moment and I could thereof produce examples but for good reasons I pass them over in silence when they think and propose by deriving new Channels from great Rivers to diminish the measure of the water in the River and to diminish it proportionally according to the measure of the Water which they make to pass through the Channel as making v.g. a Channel fifty foot broad in which the derived water is to run waste ten foot deep they think they have diminished the measure of the Water in the River five hundred feet which thing doth not indeed so fall out and the reason is plain for that the Chanel being derived the rest of the main River diminisheth in velocity and therefore retains a greater measure than it had at first before the derivation of the Chanel and moreover if the Chanel being derived it shall not conserve the same velocity which it had at first in the main River but shall diminish it it will be necessary that it hath a greater measure than it had before in the River and therefore to accompt aright there shall not be so much water derived into the Channel as shall diminish the River according to the quantity of the water in the Channel as is pretended COROLLARIE XIII THis same consideration giveth me occasion to discover a most ordinary errour observed by me in the businesse of the water of Ferara when I was in those parts in service of the most Reverend and Illustrious Monsignor Corsini the sublime wit of whom hath been a very great help to
opperations about Water as I have hitherto on sundry occasions observed are involved in so many difficulties and such a multiplicity of most extravagant accidents that it is no marvel if continually many and very important errours be therein committed by many and even by Ingeneers themselves and Learned-men and because many times they concern not onely the publique but private interests Hence it is that it not onely belongeth to Artists to treat thereof but very oft even the vulgar themselves pretend to give their judgement therein And I have been troubled many times with a necessity of treating not onely with those which either by practice or particular study understood somewhat in these matters but also with people wholly void of those notions which are necessary for one that would on good grounds discourse about this particular and thus many times have met with more difficulty in the thick skulls of men than in precipitous Torrents and vast Fennes And in particular I had occasion some years past to go see the Cave or Emissary of the Lake of Perugia made many years agon by Braccio Fortobraccio but for that it was with great ruines by Time decayed and rendred unuseful it was repaired with industry truly heroicall and admirable by Monsignor Maffei Barberino then Prefect for the Wayes and now Pope And being necessitated that I might be able to walk in the Cave and for other causes I let down the Sluices of the said Cave at the mouth of the Lake No sooner were they stopt but a great many of the people of the Towns and Villages coasting upon the Lake flocking thither began to make grievous complaints that if those Sluices were kept shut not onely the Lake would want its due Vent but also the parts adjacent to the Lake would be over flown to their very great detriment And because at first appearance their motion seemed very reasonable I found my self hard put to it seeing no way to perswade such a multitude that the prejudice which they pretended I should do them by keeping the Sluices shut for two dayes was absolutely insensible and that by keeping them open the Lake did not ebb in the same time so much as the thickness of a sheet of Paper And therefore I was necessitated to make use of the authority I had and so followed my business as cause required without any regard to that Rabble tumultuously assembled Now when I am not working with Mattock or Spade but with the Pen and Discourse I intend to demonstrate clearly to those that are capable of reason and that have well understood the ground of this my Treatise that the fear was altogether vain which those people conceited And therefore I say that the Emissary or Sluice of the Lake of Perugia standing in the same manner as at present and the water passing thorow it with the same velocity as now to examine how much the Lake may abate in two days space we ought to consider what proportion the superficies of the whole Lake hath to the measure of the Section of the Emissary and afterwards to infer that the velocity of the water by the Emissary or Sluice shall have the same proportion to the abatement of the Lake and to prove thorowly and clearly this discourse I intend to demonstrate the following Proposition That which hath been demonstrated in the Vessel falls out exactly also in our Lake of Perugia and its Emissary and because the immensity of the superficies of the Lake is in proportion to the superficies of the Emissary or Sluice as many millions to one as may be easily calculated it is manifest that such abatement shall be imperceptible and almost nothing in two dayes space nay in four or six and all this will be true when we suppose that for that time there entreth no other Water into the Lake from Ditches or Rivolets which falling into the Lake would render such abatement yet less Now we see that it 's necessary to examine such abatements and risings with excellent reasons or at least with accurate experiments before we resolve and conclude any thing and how farre the vulgar are distant from a right judgement in such matters APPENDIX XIII FOR greater confirmation of all this which I have said I will instance in another like case which also I met with heretofore wherein for that the business was not rightly understood many disorders vast expences and considerable mischiefs have followed There was heretofore an Emissary or Sluice made to drain the Waters which from Rains Springs and Rivolets fall into a Lake to the end the shores adjoyning on the Lake should be free from the overflowing of the Waters but because perhaps the enterprize was not well managed and carried on it fell out that the Fields adjacent to the said Chanel could not drain but continued under water to which disorders a present remedy hath been used namely in a time convenient to stop up the Sluice by meanes of certain Floodgates kept on purpose for that end and thus abating the Level of the Water in the Emissary in the space of three or four dayes the Fields have been haply drained But on the other part the proprietors bordering on the Lake opposed this grievously complaining that whilst the Floodgates are shut and the course of the Water of the Sluice hindered the Lake overflowes the Lands adjacent by meanes of the Rivers that fell into it to their very great damage and so continuing their suits they got more of vexation than satisfaction Now being asked my opinion herein I judged it requisite since the point in controversie was about the rising and falling of the Lake that the said abatement when the Floodgates are open and increase when they are shut should be exactly measured and told them that it might be easily done at a time when no extraordinary Waters fell into the Lake neither of Rain or otherwise and the Lake was undisturbed by winds that might drive the Water to any side by planting neer to an Islet which is about the middle of the Lake a thick post on which should be made the marks of the Lakes rising and falling for two or three dayes I would not at that time pawn or resolutely declare my judgment in regard I might be by divers accidents missed But this I told them that by what I have demonstrated and particularly that which I have said above touching the Lake of Perugia I inclined greatly to think that these risings and fallings would prove imperceptible and inconsiderable and therefore that in case experience should make good my reason it would be to no purpose for them to continue disputing and wrangling which causeth according to the Proverb A great deal of cry but produceth not much Wool Lastly it importing very much to know what a Rain continued for many dayes can do in raising these Lakes I will here insert the Copy of a Letter which I writ formerly to Signior Galilaeo Galilaei chief Philosopher to
the Grand Duke of Tuscany wherein I have delivered one of my conceits in this businesse and it may be by this Letter I may more strongly confirm what I have said above The Copy of a Letter to Signore GALILAEO GALILAEI Chief Philosopher to the most Serene Great Duke of TVSCANY Worthy and most Excellent SIR IN satisfaction of my promise in my former Letters of representing unto you some of my Considerations made upon the Lake Thrasimeno I say That in times past being in Perugia where we held our General Convention having understood that the Lake Thrasimeno by the great drought of many Moneths was much abated It came into my head to go privately and see this novelty both for my particular satisfaction as also that I might be able to relate the whole to my Patrons upon the certitude of my own sight of the place And so being come to the Emissary of the Lake I found that the Level of the Lakes surface was ebbed about five Roman Palmes of its wonted watermark insomuch that it was lower than the transome of the mouth of the Emissary by the length of this described line and therefore no Water issued out of the Lake to the great prejudice of all the places and villages circumjacent in regard that the Water which used to run from the said Lake turned 22 Mills which not going necessitated the inhabitants of those parts to go a dayes journey and more to grinde upon the Tiber. Being returned to Perugia there followed a Rain not very great but constant and even which lasted for the space of eight hours or thereabouts and it came into my thoughts to examine being in Perugia how much the Lake was increased and raised by this Rain supposing as it was probable enough that the Rain had been universal over all the Lake and like to that which fell in Perugia and to this purpose I took a Glasse formed like a Cylinder about a palme high and half a palme broad and having put in water sufficient to cover the bottome of the Glasse I noted diligently the mark of the height of the Water in the Glasse and afterwards exposed it to the open weather to receive the Raine-water which fell into it and I let it stand for the space of an hour and having observed that in that time the Water was risen in the Vessel the height of the following line I considered that if I had exposed to the same rain such other vessels equal to that the Water would have risen in them all according to that measure And thereupon concluded that also in all the whole extent of the Lake it was necessary the Water should be raised in the space of an hour the same measure Yet here I considered two difficulties that might disturb and altar such an effect or at least render it inobserveable which afterwards well weighed and resolved left me as I will tell you anon in the conclusion the more confirmed that the Lake ought to be increased in the space of eight hours that the rain lasted eight times that measure And whilst I again exposed the Glass to repeat the experiment there came unto me an Ingeneer to talk with me touching certain affairs of our Monastary of Perugia and discoursing with him I shewed him the Glass out at my Chamber window exposed in a Court-yard and communicated to him my fancy relating unto him all that I had done But I soon perceived that this brave fellow conceited me to be but of a dull brain for he smiling said unto me Sir you deceive your self I am of opinion that the Lake will not be increased by this rain so much as the thicknesse of a Julio Hearing him pronounce this his opinion with freeness and confidence I urged him to give me some reason for what he said assuring him that I would change my judgement when I saw the strength of his Arguments To which he answered that he had been very conversant about the Lake and was every day upon it and was well assured that it was not at all increased And importuning him further that he would give me some reason for his so thinking he proposed to my consideration the great drought passed and that that same rain was nothing for the great parching To which I answered I believe Sir that the surface of the Lake on which the rain had fallen was moistned and therefore saw not how its drought which was nothing at all could have drunk up any part of the rain For all this he persisting in his conceit without yielding in the least to my allegation he granted in the end I believe in civility to me that my reason was plausible and good but that in practise it could not hold At last to clear up all I made one be called and sent him to the mouth of the Emissary of the Lake with order to bring me an exact account how he found the water of the Lake in respect of the Transome of the Sluice Now here Signore Galilaeo I would not have you think that I had brought the matter in hand to concern me in my honour but believe me and there are witnesses of the same still living that my messenger returning in the evening to Perugia he brought me word that the water of the Lake began to run through the Cave and that it was risen almost a fingers breadth above the Transome Insomuch that adding this measure to that of the lowness of the surface of the Lake beneath the Transome before the rain it was manifest that the rising of the Lake caused by the rain was to a hair those four fingers breadth that I had judged it to be Two dayes after I had another bout with the Ingeneer and related to him the whole business to which he knew not what to answer Now the two difficulties which I thought of able to impede my conclusion were these following First I considered that it might be that the Wind blowing from the side where the Sluice stood to the Lake-ward the mole and mass of the Water of the Lake might be driven to the contrary shore on which the Water rising it might be fallen at the mouth of the Emissary and so the observation might be much obscured But this difficulty wholly vanished by reason of the Aires great tranquility which it kept at that time for no Wind was stirring on any side neither whilst it rained nor afterwards The second difficulty which put the rising in doubt was That having observed in Florence and elsewhere those Ponds into which the rain-water falling from the house is conveyed through the Common-shores And that they are not thereby ever filled but that they swallow all that abundance of water that runs into them by those conveyances which serve them with water insomuch that those conveyances which in time of drought maintain the Pond when there comes new abundance of water into the Pond they drink it up and swallow it A like effect might
also fall out in the Lake in which there being many veins as it is very likely that maintain and feed the Lake these veins might imbibe the new addition of the Rain-water and so by that means annull the rising or else diminish it in such sort as to render it inobservable But this difficulty was easily resolved by considering my Treatise of the measure of running-Running-Waters forasmuch as having demonstrated that the abatement of a Lake beareth the reciprocal proportion to the velocity of the Emissary which the measure of the Section of the Emissary of the Lake hath to the measure of the surface of the Lake making the calculation and account though in gross by supposing that its veins were sufficiently large and that the velocity in them were notable in drinking up the water of the Lake yet I found nevertheless that many weeks and moneths would be spent in drinking up the new-come abundance of water by the rain so that I rested sure that the rising would ensue as in effect it did And because many of accurate judgement have again caused me to question this rising setting before me that the Earth being parched by the great drought that had so long continued it might be that that Bank of Earth which environed the brink of the Lake being dry and imbibing great abundance of Water from the increasing Lake would not suffer it to increase in height I say therefore that if we would rightly consider this doubt here proposed we should in the very consideration of it see it resolved for it being supposed that that list or border of Banks which was to be occupied by the increase of the Lake be a Brace in breadth quite round the Lake and that by reason of its dryness it sucks in water and that by that means this proportion of water co-operates not in raising of the Lake It is absolutely necessary on the other hand that we consider That the Circuit of the water of the Lake being thirty miles as it s commonly held that is to say Ninety thousand Braces of Florence in compass and therefore admitting for true that each Brace of this Bank drink two quarts of water and that for the spreading it require three quarts more we shall finde that the whole agregate of this portion of water which is not imployed in the raising of the Lake will be four hundred and fifty thousand Quarts of water and supposing that the Lake be sixty square miles three thousand Braces long we shall finde that to dispence the water possest by the Bank about the Lake above the total surface of the Lake it ought to be spread so thin that one sole quart of water may over-spread ten thousand square Braces of surface such a thinness as must much exceed that of a leaf of beaten Gold and also less than that skin of water which covers the Bubbles of it and such would that be which those men would have substracted from the rising of the Lake But again in the space of a quarter of an hour at the beginning of the rain all that Bank is soaked by the said rain so that we need not for the moistning of it imploy a drop of that water which falleth into the Lake Besides we have not brought to account that abundance of water which runs in time of rain into the Lake from the steepness of the adjacent Hills and Mountains which would be enough to supply all our occasions So that neither ought we for this reason to question our pretended rising And this is what hath fallen in my way touching the consideration of the Thrasimcnian Lake After which perhaps somewhat rashly wandring beyond my bounds I proceeded to another contemplation which I will relate to you hoping that you will receive it as collected with these cautions requisite in such like affairs wherein we ought not too positively to affirm any thing of our own heads for certain but ought to submit all to the sound and secure deliberation of the Holy Mother-Church as I do this of mine and all others most ready to change my judgement and conform my self alwaies to the deliberations of my Superiors Continuing therefore my above-said conceit about the rising of the water in the glass tried before it came into my minde that the forementioned rain having been very gentle it might well be that if there should have faln a Rain fifty an hundred or a thousand times greater than this and much more intense which would insue as oft as those falling drops were four five or ten times bigger than those of the above-mentioned rain keeping the same number in such a case its manifest that in the space of an hour the Water would rise in our Glass two three and perhaps more Yards or Braces and consequently if such a Raine should fall upon a Lake that the said Lake would rise according to the same rate And likewise if such a Rain were universall over the whole Terrestriall Globe it would necessarily in the space of an hour make a rising of two or three braces round about the said Globe And because we have from Sacred Records that in the time of the Deluge it rained fourty dayes and fourty nights namely for the space of 960 houres its clear that if the said Rain had been ten times bigger than ours at Perugia the rising of the Waters above the Terrestrial Globe would reach and pass a mile higher than the tops of the Hills and Mountains that are upon the superficies of the Earth and they also would concur to increase the rise And therefore I conclude that the rise of the Waters of the Deluge have a rational congruity with natural Discourses of which I know very well that the eternal truths of the Divine leaves have no need but however I think so clear an agreement is worthy of our consideration which gives us occasion to adore and admire the greatnesse of God in his mighty Works in that we are sometimes able in some sort to measure them by the short Standard of our Reason Many Lessons also may be deduced from the same Doctrine which I passe by for that every man of himself may easily know them having once stablished this Maxime That it is not possible to pronounce any thing of a certainty touching the quantity of Running Waters by considering only the single vulgar measure of the Water without the velocity and so on the contrary he that computes only the velocity without the measure shall commit very great errours for treating of the measure of Running Waters it is necessary the water being a body in handling its quantity to consider in it all the three dimensions of breadth depth and length the two first dimensions are observed by all in the common manner and ordinary way of measuring Running Waters but the third dimension of length is omitted and haply such an oversight is committed by reason the length of Running Water is reputed in some sense infinite in that it
sense explained in the second definition SUPPOSITION III. BEcause it will often be requisite to measure the time exactly in the following Problems we take that to be an excellent way to measure the time which was shewed me many years since by Signore Galilaeo Galilaei which is as followeth A string is to be taken three Roman feet long to the end of which a Bullet of Lead is to be hanged of about two or three ounces and holding it by the other end the Plummet is to be removed from its perpendicularity a Palm more or less and then let go which will make many swings to and again passing and repassing the Perpendicular before that it stay in the same Now it being required to measure the time that is spent in any whatsoever operation those vibrations are to be numbred that are made whilst the work lasteth and they shall be so many second minutes of an hour if so be that the string be three Roman feet long but in shorter strings the vibrations are more frequent and in longer less frequent and all this still followeth whether the Plummet be little or much removed from its Perpendicularity or whether the weight of the Lead be greater or lesser These things being pre-supposed we will lay down some familiar Problems from which we shall pass to the Notions and questions more subtil and curious which will also prove profitable and not to be sleighted in this business of Waters PROPOSITION I. PROBLEME I. A Chanel of Running-Water being given the breadth of which passing through a Regulator is three Palms and the height one Palm little more or less to measure what water passeth through the Regulator in a time given FIrst we are to dam up the Chanel so that there pass not any water below the Dam then we must place in the side of the Chanel in the parts above the Regulator three or four or five Bent-pipes or Syphons according to the quantity of the water that runneth along the Chanel in such sort as that they may drink up or draw out of the Chanel all the water that the Chanel beareth and then shall we know that the Syphons drink up all the water when we see that the water at the Dam doth neither rise higher nor abate but alwaies keepeth in the same Level These things being prepared taking the Instrument to measure the time we will examine the quantity of the water that issueth by one of those Syphons in the space of twenty vibrations and the like will we do one by one with the other Syphons and then collecting the whole summe we will say that so much is the water that passeth and runneth thorow the Regulator or Chanel the Dam being taken away in the space of twenty second minutes of an hour and calculating we may easily reduce it to hours dayes months and years And it hath fallen to my turn to measure this way the waters of Mills and Fountains and I have been well assured of its exactness by often repeating the same work CONSIDERATION ANd this method must be made use of in measuring the waters that we are to bring into Conducts and carry into Cities and Castles for Fountains and that we may be able afterwards to divide and share them to particular persons justly which will prevent infinite suits and controversies that every day happen in these matters PROPOSITION II. THEOREM I. If a River moving with such a certain velocitie through its Regulator shall have a given quick height and afterwards by new water shall increase to be double it shall also increase double in velocitie LEt the quick height of a River in the Regulator ABCD be the perpendicular FB and afterwards by new water that is added to the River let the water be supposed to be raised to G so that GB may be double to EB I say that all the water GC shall be double in velocity to that of EC For the water GF having for its bed the bottom EF equally inclined as the bed BC and its quick height GE being equal to the quick height EC and having the same breadth BC it shall have of it self a velocity equal to the velocity of the first water FC but because besides its own motion which is imparted to it by the motion of the water EC it hath also over and above its own motion the motion of EC And because the two waters GC and EC are alike in velocity by the third Supposition therefore the whole water GC shall be double in velocity to the water EC which was that which we were to demonstrate This demonstration is not here inserted as perfect the Authour having by several letters to his friends confessed himself unsatisfied therewith and that he intended not to publish the Theorem without a more solid demonstration which he was in hope to light upon But being overtaken by Death he could not give the finishing touch either to this or to the rest of the second Book In consideration of which it seemed good to the Publisher of the same rather to omit it than to do any thing contrary to the mind of the Authour And this he hints by way of advertisement to those that have Manuscript Copies of this Book with the said demonstration For this time let the Reader content himself with the knowledge of so ingenious and profitable a Conclusion of the truth of which he may with small expence and much pleasure be assured by means of the experiment to be made in the same manner with that which is laid down in the second Corollary of the fourth Theorem of this with its Table and the use thereof annexed COROLLARIE HEnce it followeth that when a River increaseth in quick height by the addition of new water it also increaseth in velocity so that the velocity hath the same proportion to the velocity that the quick height hath to the quick height as may be demonstrated in the same manner PROPOS III. PROBLEME II. A Chanel of Water being given whose breadth exceeds not twenty Palms or thereabouts and whose quick height is less than five Palms to measure the quantity of the Water that runneth thorow the Chanel in a time given PLace in the Chanel a Regulator and observe the quick height in the said Regulator then let the water be turned away from the Chanel by a Chanellet of three or four Palms in breadth or thereabouts And that being done measure the quantity of the water which passeth thorow the said Chanellet as hath been taught in the second Proposition and at the same time observe exactly how much the quick height shall be abated in the greater Chanel by means of the diversion of the Chanellet and all these particulars being performed multiply the quick height of the greater Chanel into it self and likewise multiply into it self the lesser height of the said bigger Chanel and the lesser square being taken from the greater the remainder shall have the same proportion to
be abated such a measure that same shall be only one third of its whole height but moreover diverting the 4 9 it shall abate two other thirds a most principal point and such that it s not having been well understood hath caused very great disorders and there would now more than ever follow extream dammage if one should put in execution the diversion of the Sile and other Rivers and it is manifest that in the same manner wherewith it hath been demonstrated that the quantity of the Water increasing quadruple the height would increase onely double and the quantity increasing nonuple the height increaseth triple so that by adding to units all the odde numbers according to their Series the heights increase according to the natural progression of all the numbers from units As for example 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 continuing 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 Progressions of Numbers the first Series containeth all the Numbers 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 beginning at an unit and is called the Series of Quantity Heights 1 2 3 4 5 6 7 8 9 10 11 Additions 1 3 5 7 9 11 13 15 17 19 21 Quantities 1 4 9 16 25 36 49 64 81 100 121 The use of the afore-mentioned Table FIrst if we suppose the whole quick height of a River of Running Water to be divided into any number of equal parts at pleasure and would abate the same one fift by means of a division 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 abate 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 Quantities 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 2. And thus in the second place if one the contrary one would know how much water is to be added to the said River to make it increase one fift more in height so as that it may run in the Regulator 6. of those parts high of which it ran before but 5. let 6 be found in the row of heights and let the number 11. standing under the same be taken and added to the number 25. that is placed under the number 9. in the Additions and 5. in the heights and you shall have 36 which is the quantity of the water that runneth with the height of the River when it is high 6 of those parts whereof it was before but 5. 3. But if it should be desired to know how much water it is requisite to add to make the River rise so as that it may run in height 8. of those parts of which before it ran but 5 one ought to take the sum of the number of the Series of Additions standing under 8. 7. and 6 which are 15. 13. and 11. that is 39. and this shall be the summe that must be added to 25 So that to make the River to run 8. of those parts in height of which it before did run 5 it will be necessary to add 39. of those parts of which the River before was 25. 4. Likewise the same Table giveth the quantity of water that runneth from time to time through a River that increaseth by the addition of new water to the same in one of its heights the quantity of its water be known As for example If we knew that the River in one minute of an hour dischargeth 2500. of those measures of water and runneth in height 5. parts in the Regulator and afterwards should see that it runneth 8 Palms high finding in the row of quantity the number placed under 8. which is 64. we would say that the River heightned carrieth of water 64. of those parts whereof it carried before but 25 and because before it carried 2500. measures by the Golden Rule we will say that the River carrieth 6400. of those measures of which before it carried 2500. In this progress of Nature is one thing really curious and that at first sight seemeth to be somewhat Paradoxal that we proceeding ordinately in the diversions and additions with additions and diversions so unequal the abatings do notwithstanding alwaies prove equal and so do the risings And who would ever think that a River in height v. g. 10. Palms running and carrying an hundred measures in a minute of an hour is to abate but one Palm onely by the diversion of 19. of those measures and then again that the buisiness cometh to that pass that it abateth likewise a Palm by the diversion of three onely of those measures nay by the diversion of but one measure and yet it is most certain And this truth meets with so manifest proofs in experience that it is very admirable And for the full satisfaction of those who not being able to comprehend subtil demonstrations desire to be clearly inform'd by the matters of fact and to see with their bodily eyes and touch with their hands what their understanding and reason cannot reach unto I will hear add another very easie way to reduce all to an experiment the which may be made in little in great or in very great of which I make use frequently to the admiration of such as see it I prepared an hundred Siphons or if you will bowed Pipes all equal and placed them at the brim of a Vessel wherein the water is kept at one and the same level whether all the Syphons work or but a certain number of them the mouths by which the water issueth being all placed in the same level parallel to the Horizon but lower in level than the water in the Vessel and gathered all the water falling from the Syphons into another Vessel standing lower than the former I made it to run away thorow a Chanel in such manner inclined that wanting water
from the Syphons the said Chanel remained quite dry And this done I measured the quick height of the Chanel with care and afterwards divided it exactly into 10 equal parts and causing 19. of those Syphons to be taken away so that the Chanel did not run water save onely with 81 of those Syphons I again observed the quick height of the water in the same site observed before and found that its height was diminished precisely the tenth part of all its first height and thus continuing to take away 17. other Syphons the height was likewise diminished 1 10 of all its first quick height and trying to take away 15. Syphons then 13 then 11 then 9 then 7 then 5 and then 3. alwaies in these diversions made in order as hath been said there ensued still an abatement of 1 10 of the whole height And here was one thing worthy of observation that the water encreasing in or through the Chanel its quick height was different in different sites of the Chanel that is still lesser the more one approached to the Out-let notwithstanding which the abatement followed in all places proportionably that is in all its sites the first part of the height of that site diminished And moreover the water issued from the Chanel and dilated into a broader course from which likewise having divers Out-lets and Mouths yet nevertheless in that breadth also the quick heights successively varied and altered in the same proportions Nor did I here desist my observation but the water being diminished that issued from the Syphons and there being but one of them left that discharged water I observed the quick height that it made in the above-said sites the which was likewise 1 10 of all the first height there being added to the water of that Syphon the water of three other Syphons so that all the water was of 4 Syphons and consequently quadruple to the first Syphon but the quick height was onely double and adding five Siphons the quick height became triple and with adding seven Syphons the height increased quadruple and so by adding of 9. it increased quintuple and by adding of 11. it increased sextuple and by adding of 13. it increased septuple and by adding of 15. octuple and by adding of 17. nonuple and lastly by adding 19. Syphons so that all the water was centuple to the water of one Syphon yet nevertheless the quick height of all this water was onely decuple to the first height conjoyned by the water that issued from one onely Syphon For the more clear understanding of all which I have made the following Figure in which we have the mouth A that maintaineth the water of the Vessel BC in the same level though it continually run to the brim of the Vessel are put 25. Syphons and there may be many more divided into 5 Classes DEFGH and the first D are of one onely Syphon the second E of three Syphons the third F of five the fourth G of 7 the fifth H of 9 and one may suppose the sixth of 11 the seventh of 13 Syphons and so of the other Classes all containing in consequent odd numbers successively we are content to represent in the Figure no more but the five forenamed Classes to avoid confusion the gathered water DEFGH which runneth thorow the Chanel IKL and falleth into the out-let MNOP and so much sufficeth for the explanation of this experiment PROPOS V. PROB. III. Any River of any bigness if being given to examine the quantity of the Water that runneth thorow the River in a time assigned BY what we have said already in the two preceding Problems we may also resolve this that we have now before us and it is done by diverting in the first place from the great River a good big measurable Chanel as is taught in the second Probleme and observing the abatement of the River caused by the diversion of the Chanel and finding the proportion that the Water of the Chanel hath to that of the River then let the Water of the Chanel be measured by the second Probleme and work as above and you shall have your desire CONSIDERATION I. ANd although it seemeth as if it might prove difficult and almost impossible to make use of the Regulator number if one be about to measure the water of some great River and consequently would be impossible or at least very difficult to reduce the Theory of the first Probleme into practice Yet nevertheless I could say that such great conceits of measuring the water of a great River are not to come into the minds of any but great Personages and potent Princes of whom it is expected for their extraordinary concerns that they will make these kinde of enquiries as if here in Italy it should be of the Rivers Tyber Velino Chiana Arno Serchio Adice in which it seemeth really difficult to apply the Regulator to finde exactly the quick height of the River But because in such like cases sometimes it would turn to account to be at some charge to come to the exact and true knowledge of the quantity of water which that River carrieth by knowledge whereof other greater disbursments might afterwards be avoided that would oft times be made in vain and prevent the disgusts which sometimes happen amongst Princes Upon this ground I think it will be well to shew also the way how to make use of the Regulator in these great Rivers in which if we will but open our eyes we shall meet with good ones and those made without great cost or labour which will serve our turn For upon such like Rivers there are Wears or Lockes made to cause the Waters to rise and to turn them for the service of Mills or the like Now in these Cases it is sufficient that one erect upon the two extreames of the Weare two Pilasters either of Wood or Brick which with the bottome of the Weare do compose our Regulator wherewith we may make our desired operation yea the Chanel it self diverted shall serve without making any other diversion or union And in brief if the businesses be but managed by a judicious person there may wayes and helps be made use of according to occasion of which it would be too tedious to speak and therefore this little that hath been hinted shall suffice CONSIDERATION II. FRom what hath been declared if it shall be well understood may be deduced many benefits and conveniences not onely in dividing of Running Waters for infinite uses that they are put to in turning of Corne-Mills Paper-Mills Gins Powder-Mills Rice-Mills Iron-Mills Oil-Mills Sawing-Mills Mirtle-Mills Felling-Mills Fulling-Mills Silk-Mills and such other Machines but also in ordering Navigable Chanels diverting Rivers and Chanels of Waters or terminating and limiting the sizes of Pipes for Fountains In all which affairs there are great errours comâ—â—â—ed to the losse of much expence the Chanels and Pipes that are made sometimes not being sufficient to carry the designed Waters and sometimes they
are made bigger than is necessary which disorders shall be avoided if the Engineer be advised of the things abovesaid and in case that to these Notions there be added the knowledge of Philosophy and Mathematicks agreeable to the sublime Discoveries of Signore Galilaeo and the further improvement thereof by Signore Evangelista Torricelli Mathematician to the Grand Duke of Tuscany who hath subtilly and admirably handled this whole businesse of Motion one shall then come to the knowledge of particular notions of great curiosity in the Theoricks and of extraordinary benefit in the Practicks that daily occur in these businesses And to shew in effect of what utility these Notions are I have thought fit to insert in this place the Considerations by me made upon the Lake of Venice and to represent at large by the experience of the last year 1641. the most Serene Erizzo then Duke of the said Republique Being therefore at Venice in the year aforesaid I was requested by the most Illustrious and most Excellent Signore Giovanni Basadonna a Senatour of great worth and merit that I would ingenously deliver my opinion touching the state of the Lake of Venice and after I had discoursed with his Honour several times in the end I had order to set down the whole businesse in writing who having afterwards read it privately the said Signore imparted the same with like privacy to the most Serene PRINCE and I received order to represent the same to the full Colledge as accordingly I did in the Moneth of May the same year and it was as followeth CONSIDERATIONS Concerning the LAKE OF VENICE BY D. BENEDETTO CASTELLI Abbot of S. Benedetto Aloysio Mathematician to Pope VRBAN VIII and Professor in ROME CONSIDERATION I. THough the principal cause be but one onely that in my judgment threatneth irreparable ruine to the Lake of Venice in the present state in which it now stands Yet neverthelesse I think that two Heads may be considered And this Consideration may peradventure serve us for to facilitate and explain the opportune remedies though not to render the state of things absolutely unchangeable and eternal an enterprize impossible and especially in that which having had some beginning ought likewise necessarily to have its end or at least to prevent the danger for many hundreds of years and possibly it may in the mean time by the mutation it self be brought into a better condition I say therefore that the present disorder may be considered under two Heads One is the very notable discovery of Land that is observed at the time of low Water the which besides the obstructing of Navigation in the Lake and also in the Chanels doth likewise threaten another mischief and disorder worthy of very particular consideration which is That the Sun drying up that mudde especially in the times of hot Summers doth raise thence the putrified and pernicious vapours fogs and exhalations that infect the Air and may render the City unhabitable The second Head is the great Stoppage that daily is growing in the Ports especially of Venice at Malamoco concerning which matters I will hint certain general points and then will proceed to the more particular and important affairs And first I say that I hold it altogether impossible to effect any thing though never so profitable which doth not bring with it some mischief and therefore the good and the hurt ought to be very well weighed and then the lesse harmful part to be imbraced Secondly I propose to consideration that the so notable discovery of Earth Mud hath not been long observed as I understand from old persons that can remember passages for fifty years past which thing being true as to me it seemeth most true it should appear that it could not but be good to reduce matters to that passe that they were at formerly laying aside all affection or passion that self-flattering minds have entertained for their own conceits or at least it shall be necessary speedily to consult the whole Thirdly I hold that it is necessary to weigh whether from the foresaid discovery of Land it followeth that onely the Earth riseth as it is commonly thought by all without dispute or whether the Waters are abated and faln away or else whether it proceedeth from both the one and other cause And here it would be seasonable to enquire what share the said causes may have each considered apart in the foresaid effect For in the first case if the Earth have been raised it would be necessary to consider of taking it down and removing it But if the Waters have failed or abated I believe that it would be extreamly necessary to restore and raise them And if both these reasons have conspired in this effect it will be necessary to remedy them each apart And I do for my part think that the so notable appearance of Shelves at the time of low Water proceeds principally from the decrease and abatement of the Waters which may confidently be affirmed to need no other proof in regard that the Brent hath been actually diverted which did formerly discharge its Water into the Lake As to the other point of the great Stoppage of Ports I hold that all proceedeth from the violence of the Sea which being sometimes disturbed by windes especially at the time of the waters flowing doth continually raise from its bottome immense heaps of sand carrying them by the tide and force of the waves into the Lake it not having on its part any sttength of current that may raise and carry them away they sink to the bottom and so they choke up the Ports And that this effect happeneth in this manner we have most frequent experiences thereof along the Sea-coasts And I have observed in Tuscany on the Roman-shores and in the Kingdom of of Naples that when a river falleth into the Sea there is alwaies seen in the Sea it self at the place of the rivers out-let the resemblance as it were of an half-Moon or a great shelf of settled sand under water much higher then the rest of the shore and it is called in Tuscany il Cavallo and here in Venice lo Sâ—anto the which cometh to be cut by the current of the river one while on the right side another while on the left and sometimes in the midst according as the Wind sits And a like effect I have observed in certain little Rillets of water along the Lake of Bolsena with no other difference save that of small and great Now whoso well considereth this effect plainly seeth that it proceeds from no other than from the contrariety of the stream of the River to the impetus of the Sea-waves seeing that great abundance of sand which the Sea continually throws upon the shore cometh to be driven into the Sea by the stream of the river and in that place where those two impediments meet with equal force the sand setleth under water and thereupon is made that same Shelf for Cavallo the which if the
prove impassible for Navigation with a manifest danger of corrupting the Air And in the last place there will unavoidably ensue the choaking and stoppage of the Ports of Venice Upon the 20th of December 1641. I imparted this my second Consideration to the most Excellent Signore Basadonna presenting him with a Copy thereof amongst other Writings which I have thought good to insert although they seem not to belong directly to our businesse of the Lake The way to examine the MUD and SAND that entereth and remaineth in the LAKE OF VENICE To the most Excellent SIGNORE GIO. BASADONNA TWo very considerable Objections have been made against my opinion concerning the Lake of Venice One was that of which I have spoken at large in my first Consideration namely that the Brents having been taken out of the Lake cannot have been the occasion of the notable fall of the Waters in the Lake as I pretend and consequently that the turning Brent into the Lake would be no considerable remedy in regard that the water of Brent and the great expansion of the Lake over which the water of Brent is to diffuse and spread being considered it is found that the rise proveth insensible The second Objection was that the Brent is very muddy and therefore if it should fall muddy into the Lake the Sand would sink and fill up the same Touching the first Query enough hath been said in my first Consideration where I have plainly discovered the deceipt of the Argument and shewn its fallacy It remaineth now to examine the second to which in the first place I say that one of the first things that I proposed in this affair was that I held it impossible to do any act though never so beneficial that was not also accompanied by some inconvenience and mischief and therefore we are to consider well the profit and the losse and prejudice and they both being weighed we shall be able to choose the lesser evil Secondly I admit it to be most true that Brent is at some times muddy but it is also true that for the greater part of the year it is not muddy Thirdly I do not see nor understand what strength this objection hath being taken so at large and in general and methinks that it is not enough to say that the Brent runneth muddy and to assert that it deposeth its Muddinesse in the Lake but we ought moreover to proceed to particulars and shew how much this Mud is and in what time this choaking up of the Ports may be effected For the Reasons are but too apparent and particular that conclude the ruine of the Lake and that in a very short time for mention is made of dayes the Waters diversion being made and moreover we have the circumstance of an Experiment the state of things being observed to have grown worse since the said diversion And I have demonstrated that in case the Diversion of the Sile and the other Rivers should be put in execution the Lake would in a few dayes become almost dry and the Ports would be lost with other mischievous consequences But on the other side although that we did grant the choaking of them we may very probably say that it will not happen save onely in the succession of many and many Centuries of years Nor can I think it prudent counsel to take a resolution and imbrace a Designe now to obtain a benefit very uncertain and more than that which only shall concern those who are to come very many Ages after us and thereby bring a cetain inconvenience upon our selves and upon our children that are now alive and present Let it be alledged therefore although I hold it false that by the diversions of the Rivers the Lake may be kept in good condition for several years to come But I say confidently and hope to demonstrate it That the Diversions will bring the Lake even in our dayes to be almost dry and at least will leave so little water in it that it shall cease to be Navigable and the Ports shall most infallibly be choaked up I will therefore say upon experience in answer to this Objection that it is very necessary first well to discourse and rationally to particularize and ascertain the best that may be this point of the quantity of this sinking Mud or Sand. Now I fear I shall make my self ridiculous to those who measuring the things of Nature with the shallownesse of their brains do think that it is absolutely impossible to make this enquiry and will say unto me Quis mensus est pugillo aquas terram palmo ponderavit Yet nevertheless I will propound a way whereby at least in gross one may find out the same Take a Vessel of Cylindrical Figure holding two barrels of water or thereabouts and then fill it with the water of Brent at its Mouth or Fall into the Lake but in the Lake at the time that the Brent runneth muddy and after it hath begun to run muddy for eight or ten hours to give the mud time to go as far as S. Nicolo to issue into the Sea and at the same time take another Vessel like and equal to the first and fill it with the water of the Lake towards S. Nicolo but take notice that this operation ought to be made at the time when the waters go out and when the Sea is calm and then when the waters shall have setled in the aforesaid Vessels take out the clear water and consider the quantity of Sand that remains behind and let it be set down or kept in mind And I am easily induced to think that that shall be a greater quantity of Sand which shall be left in the first Vessel than that left in the second Vessel Afterwards when the Brent shall come to be clear let both the operations be repeated and observe the quantity of Sand in the aforesaid Vessels for if the Sand in the first Vessel should be most it would be a sign that in the revolution of a year the Brent would depose Sand in the Lake And in this manner one may calculate to a small matter what proportion the Sand that entreth into the Lake hath to that which remains And by that proportion one may judge how expedient it shall be for publick benefit And if at several times of the year you carefully repeat the same operations or rather observations you would come to a more exact knowledge in this business And it would be good to make the said operations at those times when the Lake is disturbed by strong high Winds and made muddy by its own Mud raised by the commotion of the Waters This notion would give us great light if the same observations should be made towards the Mouth of Lio at such time as the waters flow and ebb in calm seasons for so one should come to know whether the waters of the Lake are more thick at the going out than at the entrance I have propounded the