SALV. Fair and &longs;oftly, Simplicius, your &longs;cruple is not &longs;o ab­&longs;truce, nor I &longs;o incautelous, that you &longs;hould need to think that I was not advi&longs;ed of it, and that con&longs;equently I have not found a re­ply to it. Therefore, for my explanation, and your information, hearken to what I &longs;hall &longs;ay. We are upon the examination of what would befall Moveables exceeding different in weight in a Medium, in ca&longs;e it &longs;hould have no Re&longs;i&longs;tance, &longs;o that all the diffe­rence of Velocity that is found between the &longs;aid Moveables ought to be referred to the &longs;ole inequality of Weight. And becau&longs;e on­ly a Space altogether void of Air, and of every other, though te­nuous and yielding Body, would be apt &longs;en&longs;ibly to &longs;hew us what we &longs;eek, &longs;ince we want &longs;uch a Space, let us &longs;ucce&longs;&longs;ively ob&longs;erve that which happeneth in the more &longs;ubtill and le&longs;&longs;e re&longs;i&longs;ting Mediums,in compari&longs;on of that which we &longs;ee to happen in others le&longs;&longs;e &longs;ubtill and more re&longs;i&longs;ting: for if we &longs;hould really find the Moveables different in Gravity to differ le&longs;&longs;e and le&longs;&longs;e in Velocity, according as the Mediums are found more and more yielding; and that, finally, although extreamly unequal in weight, in a Medium more tenuous than any other, though not void, the difference of Velo­city di&longs;covers it &longs;elf to be very &longs;mall, and almo&longs;t unob&longs;ervable, I conceive that we may, and that upon very probable conjecture, believe, that in a Vacuum their Velocities would be exactly equal. Therefore let us con&longs;ider that which hapneth in the Air; wherein to have a Figure of an uniform Superficies, and very light Matter, I will that we take a blown Bladder, in which the included Air will weigh little or nothing in a Medium of the Air it &longs;elf, becau&longs;e it can make but very &longs;mall Compre&longs;&longs;ion therein, &longs;o that the Gravi­ty is only that little of the &longs;aid film, which would not be the thou­&longs;andth part of the weight of a lump of Lead of the bigne&longs;s of the &longs;aid Bladder when blown. The&longs;e, Simplicius, being let fall from the height of four or &longs;ix yards, how great a &longs;pace, do you judge, that the Lead would anticipate the Bladder in its de&longs;cent? A&longs;&longs;ure your &longs;elf that would not move thrice, no nor twice as fa&longs;t, although even now you would have had it to have been a thou­&longs;and times more &longs;wift.

SIMP. It is po&longs;&longs;ible that at the beginning of the Motion, that is, in the fir&longs;t five or &longs;ix yards this might happen that you &longs;ay; but in the progre&longs;&longs;e, and in a long continuation I believe, that the Lead would leave it behind, not only &longs;ix, but al&longs;o eight and ten parts of twelve.

SALV. And I al&longs;o believe the &longs;ame: and make no que&longs;tion, but that in very great di&longs;tances the Lead will have pa&longs;&longs;ed an hun­dred miles of way, ere the Bladder will have pa&longs;&longs;ed &longs;o much as one. But this, Simplicius, which you propound, as an effect contrary to my A&longs;&longs;ertion, is that which mo&longs;t e&longs;pecially confirmeth it. It is (I once more tell you) my intent to declare, That the difference of Gravity is in no wi&longs;e the cau&longs;e of the divers velocities of Movea­bles of different Gravity, but that the &longs;ame dependeth on exteri­our accidents, & in particular, on the Re&longs;i&longs;tance of the Medium, &longs;o that, this being removed, all Moveables move with the &longs;ame de­grees of Velocity. And this I chiefly deduce from that which but now you your &longs;elf did admit, and which is very true, namely, that of two Moveables, very different in weight, the Velocities more and more differ, according as the ^{*} Spaces are greater and greater that they pa&longs;&longs;e: an Effect which would not follow, if it did depend on the different Gravities: for they being alwaies the &longs;ame, the pro­portion betwixt the Spaces would likewi&longs;e alwaies continue the &longs;ame, which proportion we &longs;ee &longs;till &longs;ucce&longs;&longs;ively to encrea&longs;e in the continuance of the Motion; for that the heavie&longs;t Moveable in the de&longs;cent of one yard will not anticipate the lighte&longs;t the tenth part of that Space or Way, but in the fall of twelve yards will out-go it a third part, in that of an hundred will out&longs;trip it 90/100.

* Or Waies.

SIMP. Very well: But following you &longs;tep by &longs;tep, if the dif­ference of weight in Moveables of different Gravities cannot cau&longs;e the difference of proportion in their Velocities, for that the Gravities do not alter; neither then can the Medium, which is &longs;uppo&longs;ed alwaies to continue the &longs;ame, cau&longs;e any alteration in the proportion of the Velocities.

SALV. You wittily bring an in&longs;tance again&longs;t my Po&longs;ition, that it is very nece&longs;&longs;ary to remove. I &longs;ay therefore, that a Grave Body hath, by Nature, an intrin&longs;ick Principle of moving towards the Common Center of heavy things, that is to that of our Terre&longs;trial Globe, with a Motion continually accelerated, and accelerated alwaies equally, &longs;cilicet, that in equal times there are made equal ^{*} additions of new Moments, and degrees of Velocities: and this ought to be under&longs;tood to hold true at all times when all acciden­tal and external impediments are removed; among&longs;t which there is one that we cannot obviate, that is the Impediment of the Me­dium, which is Repleat, when as it &longs;hould be opened and latterally moved by the falling Moveable, to which tran&longs;ver&longs;e Motion the Medium, though fluid, yielding and tranquile, oppo&longs;eth it &longs;elf with a Re&longs;i&longs;tance one while le&longs;&longs;er, and another while greater and greater, according as it is more &longs;lowly or ha&longs;tily to open to give pa&longs;&longs;age to the Moveable, which, becau&longs;e, as I have &longs;aid, it goeth of its own nature continually accelerating, it cometh of con&longs;e­quence to encounter continually greater Re&longs;i&longs;tance in the Medi­um, and therefore Retardment, and diminution in the acqui&longs;t of new degrees of Velocity; &longs;o that in the end, the Velocity arriveth to that &longs;wiftne&longs;&longs;e, and the Re&longs;i&longs;tance of the Medium, to that &longs;trength, that ballancing each other, they take away all further Acceleration, and reduce the Moveable to an Equable and Uni­form Motion, in which it afterwards continually abides. There is therefore in the Medium augmentation of Re&longs;i&longs;tance, not becau&longs;e it changeth its E&longs;&longs;ence, but becau&longs;e the Velocity altereth where­with it ought to open, and laterally move, to give pa&longs;&longs;age to the falling Body, which goeth continually accelerating. Now the ob&longs;erving, that the Re&longs;i&longs;tance of the Air to the &longs;mall Moment or Impetus of the Bladder is very great, and to the great weight of the Lead is very &longs;mall, makes me hold for certain, that if one &longs;hould wholly remove it, by adding to the Bladder great a&longs;&longs;i&longs;tance, and but very little to the Lead, their Velocities would equalize each other. Taking this Principle therefore for granted, That in the Medium wherein, either by rea&longs;on of Vacuity, or otherwi&longs;e, there were no Re&longs;i&longs;tance that might abate the Velocity of the Motion, &longs;o that of all Moveables the Velocities were alike, we might con­gruou&longs;ly enough a&longs;&longs;ign the proportions of the Velocities of like and unlike Moveables, in the &longs;ame and in different, Replear, and therefore Re&longs;i&longs;ting Medium's. And this we might effect by &longs;tudy­ing how much the Gravity of the Medium abateth from the Gra­vity of the Moveable, which Gravity is the In&longs;trument wherewith the Moveable makes its Way, repelling the parts of the Mediumon each Side: an operation that doth not occur in void Mediums; and therefore there is no difference to be expected from the di­ver&longs;e Gravity: and becau&longs;e it is manife&longs;t, that the Medium abateth from the Gravity of the Body by it contained, as much as is the weight of &longs;uch another ma&longs;s of its own Matter, if the Velocities of the Moveables that in a non-re&longs;i&longs;ting Medium would be (as hath been &longs;uppo&longs;ed) equal, &longs;hould dimini&longs;h in that proportion, we &longs;hould have what we de&longs;ired. As for example; &longs;uppo&longs;ing that Lead be ten thou&longs;and times more grave than Air, but Ebony a thou&longs;and times only; of the Velocities of the&longs;e two Matters, which ab&longs;olutely taken, that is, all Re&longs;i&longs;tance being removed, would be equal, the Air &longs;ub&longs;tracts from the ten thou&longs;and degrees of the Lead one, and from the thou&longs;and degrees of the Ebony likewi&longs;e abateth one, or, if you will, of its ten thou&longs;and, ten. If there­fore the Lead and the Ebony &longs;hall de&longs;cend thorow the Air from any height, which, the retardment of the Air removed, they would have pa&longs;&longs;ed in the &longs;ame time, the Air will abate from the ten thou&longs;and degrees of the Leads Velocity one, but from the ten thou&longs;and degrees of Ebony's Velocity it will abate ten: which is as much as to &longs;ay, that dividing that Altitude, from which tho&longs;e Moveables departed into ten thou&longs;and parts, the Lead will arrive at the Earth, the Ebony being left behind, ten, nay, nine of tho&longs;e &longs;ame ten thou&longs;and parts. And what el&longs;e is this, but that a Ball of Lead, falling from a Tower two hundred yards high, to find how much it will anticipate one of Ebony of le&longs;&longs;e than four Inches? The Ebony weigheth a thou&longs;and times more than the Air, but that Bladder &longs;o blown, weigheth only four times &longs;o much; the Air therefore from the intrin&longs;ick and natural Velocity of the Ebony &longs;ubducteth one degree of a thou&longs;and, but from that, which al&longs;o in the Bladder would ab&longs;olutely have been the &longs;ame, the Air &longs;ub­ducts one part of four: &longs;o that by that time the Ball of Ebony falling from the Tower, &longs;hall come to the ground, the Bladder &longs;hall have pa&longs;&longs;ed but three quarters of that height. Lead is twelve times heavier than Water, but Ivory only twice as heavy; the Water therefore, from their ab&longs;olute Velocities which would be equal, &longs;hall abate in the Lead the twelfth part, but in the Ivory the half: when therefore, in the Water, the Lead &longs;hall have de­&longs;cended eleven fathom, the Ivory &longs;hall have de&longs;cended &longs;ix. And, arguing by this Rule, I believe, that we &longs;hall find the Experiment much more exactly agree with this &longs;ame Computation, than with that of Ari&longs;totle. By the like method we might find the Veloci­ties of the &longs;ame Moveable in different fluid Mediums, not compa­ring the different Re&longs;i&longs;tances of the Mediums, but con&longs;idering the exce&longs;&longs;es of the Gravity of the Moveable over and above the Gra­vities of the Mediums: v. gr. ^{*} Tin is a thou&longs;and times heavier than Air, and ten times heavier than Water; therefore dividing the ab­&longs;olute Velocity of the Tin into a thou&longs;and degrees, it &longs;hall move in the Air, (which deducteth from it the thou&longs;andth part,) with nine hundred ninety nine, but in the Water with nine hundred only; being that the Water abateth the tenth part of its Gravity, and the Air the thou&longs;andth part. Take a Solid &longs;omewhat heavier than Water, as for in&longs;tance, the Wood called Oake, a Ball of which weighing, as we will &longs;uppo&longs;e, a thou&longs;and drams, a like quantity of Water will weigh nine hundred and fifty, but &longs;o much Air will weigh but two drams,: it is manife&longs;t, that &longs;uppo&longs;ing that its ab&longs;o­lute Velocity were of a thou&longs;and degrees, in Air there would re­main nine hundred ninety eight, but in the Water only fifty; be­cau&longs;e that the Water of the thou&longs;and degrees of Gravity taketh away nine hundred and fifty, and leaves fifty only; that Solid there­fore would move well-near twenty times as fa&longs;t in the Air as Wa­ter; like as the exce&longs;&longs;e of its Gravity above that of the Water is the twentieth part of its own. And here I de&longs;ire that we may con­&longs;ider, that no matters, having a power to move downwards in the Water, but &longs;uch as are more grave in Species than it; and con&longs;e­quently many hundreds of times, more grave than the Air, in &longs;eeking what the proportions of their Velocities are in the Air and Water, we may, without any con&longs;iderable errour, make account that the Air doth not deduct any thing of moment from the ab&longs;o­lute Gravity, and con&longs;equently, from the ab&longs;olute Velocity of &longs;uch matters: &longs;o that having ea&longs;ily found the exce&longs;&longs;e of their Gravi­ty above the Gravity of the Water, we may &longs;ay that their Velo­city in the Air, to their Velocity in the Water hath the &longs;ame propor­tion, that their total Gravity hath to the exce&longs;&longs;e of this above the Gravity of the Water. For example, a Ball of Ivory weigh­eth twenty ounces, a like quantity of Water weigheth &longs;eventeen ounces: therefore the Velocity of the Ivory in Air, to its Velocity in Water is very neer as twenty to three.

The Velocity of Grave Bodies de­&longs;cending Natural­ly to the Center do go continually en­crea&longs;ing till that by the encrea&longs;e of the Re&longs;i&longs;tance of the Medium it becometh uniform.

* Or aqui&longs;ts.

To find the Pro­portions of the Ve­locities of different Moveables in the &longs;ame, and in diffe­rent Mediums.

* Or Pewter.

SAGR. I have made a great acqui&longs;t in a bu&longs;ine&longs;&longs;e of it &longs;elf cu­rious, and in which, but without any benefit, I have many times wearied my-thoughts: nor would there any thing be wanting for the putting the&longs;e Speculations in practice, &longs;ave onely the way how one &longs;hould come to know of what Gravity the Air, is in com­pari&longs;on to the Water, and con&longs;equently to other heavy matters.

SIMP. But in ca&longs;e one &longs;hould finde, that the Air in&longs;tead of Gravity had Levity, what ought one to &longs;ay of the foregoing di&longs;­cour&longs;es, otherwi&longs;e very ingenuous?

SALV. It would be nece&longs;&longs;ary to confe&longs;&longs;e that they were truly Aerial, Light, and Vain. But will you que&longs;tion whether the Air be heavy, having the expre&longs;&longs;e Text of Ari&longs;totle that affirmeth it, &longs;aying, That all the Elements have Gravity, even the Air it &longs;elf; a &longs;igne of which (&longs;ubjoyns he) we have in that a ^{*} Bladder blown, weigheth heavier than un&longs;well'd.

* Or Boracho; a bottle made of a Goat skin, u&longs;ed to hold wine and other Liquids.

SIMP. That a Boracho, or Bladder blown, weigheth more, might proceed, as I could &longs;uppo&longs;e, not from the Gravity that is in the Air, but in the many gro&longs;&longs;e Vapours intermixed with it in the&longs;e our lower Regions; by means whereof I might &longs;ay, that the Gravity of the Bladder, or Boracho encrea&longs;eth.

SALV. I would not have you &longs;ay it, and much le&longs;&longs;e that you &longs;hould make Aristotle &longs;peak it, for he treating of the Elements, and de&longs;iring to per&longs;wade me that the Element of Air is grave, making me to &longs;ee it by an Experement: if in comming to the proof he &longs;hould &longs;ay: Take a Bladder, and fill it with gro&longs;&longs;e Vapours; and ob&longs;erve that its weight will encrea&longs;e; I would tell him that it would weigh yet more if one &longs;hould fill it with bran; but would afterwards adde; that tho&longs;e Experiments prove, that bran, and gro&longs;&longs;e Vapours are grave: but as to the Element of Air, I &longs;hould be left in the &longs;ame doubt as before. The Experiment of Ari&longs;totletherefore is good, and the Propo&longs;ition true. But I will not &longs;ay &longs;o much, for a certain other rea&longs;on taken expre&longs;ly out of a Philo&longs;o­pher who&longs;e name I do not remember, but am &longs;ure that I have read it, who argueth the Air to be more grave than light, becau&longs;e it more ea&longs;ily carrieth grave Bodies downwards, than the light up­wards.

SAGR. Good i-faith. By this rea&longs;on then, the Air &longs;hall be much heavier than the Water, &longs;ince, that all Bodies are carried more ea&longs;ily downwards thorow the Air than thorow the Water, and all light Bodies more ea&longs;ily upwards in this than in that: nay, infinite matters a&longs;cend in the Water, that in the Air de&longs;cend. But be the Gravity of the Bladder, Simplicius, either by rea&longs;on of the gro&longs;&longs;e Vapours, or pure Air, this nothing concerns our pur­po&longs;e, for we &longs;eek that which happeneth to Moveables that move in this our Vaporous Region. Therefore, returning to that which more concerneth me, I would for a full and ab&longs;olute informati­on in the pre&longs;ent bu&longs;ine&longs;&longs;e, not onely be a&longs;&longs;ured that the Air is grave, as I hold for certain, but I would, if it be po&longs;&longs;ible, know what its Gravity is. Therefore, Salviatus, if you have wherewith to &longs;atisfie me in this al&longs;o, I entreat you to favour me with the &longs;ame.

SALV. That there re&longs;ideth in the Air po&longs;itive Gravity, and not, as &longs;ome have thought, Levity, which haply is in no Mat­ter to be found, the Experiment of the Blown-Bladder, alledged by Ari&longs;totle, affordeth us a &longs;ufficiently-convincing Argument; for if the quality of ab&longs;olute and po&longs;itive Levity were in the Air, then the Air being multiplied and compre&longs;&longs;ed, the Levity would encrea&longs;e, and con&longs;equently the propen&longs;ion of going upwards: but Experience &longs;hews the contrary. As to the other demand, that is, of the Method how to inve&longs;tigate its Gravity, I have tried to do it in this manner: I have taken a pretty bigge Gla&longs;&longs;e ^{*} Bottle, with its neck bended, and a Finger-&longs;tall of Leather fa&longs;t about it, having in the top of the &longs;aid Finger-&longs;tall in&longs;erted and fa­&longs;tened a Valve of Leather, by which with a Siringe I have made pa&longs;&longs;e into the Bottle by force a great quantity of Air, of which, becau&longs;e it admits of great Conden&longs;ation, it may take in two or three other Bottles-ful over and above that which is naturally con­tained therein. Then I have in an exact Ballance very preci&longs;ely weighed that Bottle with the Air compre&longs;&longs;ed within it, adju&longs;ting the weight with &longs;mall Sands. Afterwards, the Valve being opened, and the Air let out, that was violently conteined in the Ve&longs;&longs;el, I have put it again into the Scales, and finding it notably aleviated, I have by degrees taken &longs;o much Sand from the other Scale, keep­ing it by it &longs;elf, that the Ballance hath at la&longs;t &longs;tood in Equilibriowith the remaining counter-poi&longs;e, that is with the Bottle. And here there is no que&longs;tion, but that the weight of the re&longs;erved Sand is that of the Air that was forceably driven into the Bottle, and which is at la&longs;t gone out thence. But this Experiment hitherto a&longs;­&longs;ureth me of no more but this, that the Air violently deteined in the Ve&longs;&longs;el, weigheth as much as the re&longs;erved Sand, but how much the Air re&longs;olutely and determinately weigheth in re&longs;pect of the Water, or other grave matter, I do not as yet know, nor can I tell, unle&longs;&longs;e I mea&longs;ure the quantity of the Air compre&longs;&longs;ed: and for the di&longs;covering of this a Rule is nece&longs;&longs;ary, which I have found may be performed two manner of wayes, one of which is to take &longs;uch another Bottle or Flask as the former, and in like manner bended, with a Finger-&longs;tall of Leather, the end of which may clo&longs;ely imbrace the Volve of the other, and let it be very fa&longs;t tied about it. It's requi&longs;ite, that this &longs;econd Bottle be bored in the bottom, &longs;o that as by that hole we may thru&longs;t in a Wier, wherewith we may, at plea&longs;ure, open the &longs;aid Volve, to let out the &longs;uperfluous Air of the other Ve&longs;&longs;el, after it hath been weighed: but this &longs;econd Bottle ought to be full of Water. All being pre­pared in the manner afore&longs;aid, and with the Wier opening the Volve, the Air i&longs;&longs;uing out with impetuo&longs;ity, and pa&longs;&longs;ing into the Ve&longs;&longs;el of Water, &longs;hall drive it out by the hole at the Bottom: and it is manife&longs;t, that the quantity of Water which &longs;hall be thru&longs;t out, is equal to the Ma&longs;&longs;e and quantity of Air that &longs;hall have i&longs;&longs;ued from th'other Ve&longs;&longs;el: that Water therefore being kept, and returning to weigh the Ve&longs;&longs;el lightned of the Air com­pre&longs;&longs;ed (which I &longs;uppo&longs;e to have been weighed likewi&longs;e fir&longs;t with the &longs;aid forced Air) and the &longs;uperfluous &longs;and being laid by, as I directed before; it is manife&longs;t, that this is the ju&longs;t weight of &longs;o much Air in ma&longs;&longs;e, as is the ma&longs;&longs;e of the expul&longs;ed and re&longs;erved Water; which we are to weigh, and &longs;ee how many times its weight &longs;hall contain the weight of the re&longs;erved &longs;and: and we may without errour affirme, that the Water is &longs;o many times heavier than Air; which &longs;hall not be ten times, as it &longs;eemeth Ari&longs;totleheld, but very neer four hundred, as the &longs;aid Experiment &longs;heweth.

The Air hath Po­&longs;itive Gravity.

How that Gravity may be computed.

* Un Fia&longs;co, tho&longs;e long-neckt gla&longs;&longs;e bottles in which we have our Florence Wine brought to us.

The other way is more expeditious, and it may be done with one Ve&longs;&longs;el onely, that is with the fir&longs;t accomodated after the man­ner before directed, into which I will not that any other Air be put, more than that which naturally is found therein; but I will, that we inject Water without &longs;uffering any Air to come out, which being forced to yield to the &longs;upervenient Water mu&longs;t of nece&longs;&longs;ity be compre&longs;&longs;ed: having gotten in, therefore, as much Water as is po&longs;&longs;ible, (but yet without great violence one cannot get in three quarters of what the Bottle will hold) put it into the Scales, and very carefully weigh it: which done, holding the Ve&longs;&longs;el with the neck upwards, open the Volve, letting out the Air, of which there will preci&longs;ely i&longs;&longs;ue forth &longs;o much as there is Water in the Bottle. The Air being gone out, put the Ve&longs;&longs;el again into the Scales, which by the departure of the Air will be found lightened, and abating from the oppo&longs;ite Scale the &longs;uperfluous weight, it &longs;hall give us the weight of as much Air as there is Water in the Bottle.

SIMP. The Contrivances you found out cannot but be con­fe&longs;&longs;ed to be witty and very ingenuous, but whil&longs;t, me thinks, they fully &longs;atisfie my under&longs;tanding, they another way occa&longs;ion in me much Confu&longs;ion, for it being undoubtedly true that the Ele­ments in their proper Region are neither heavy nor light, I can­not comprehend, how and which way that portion of Air, which &longs;eemeth to have weighed v. gr. four drams of &longs;and, &longs;hould af­terwards have that &longs;ame Gravity in the Air, in which the &longs;and is contained that weigheth again&longs;t it: and therefore me thinks that the Experiment ought not to be practiced in the Element of Air, but in a Medium in which the Air it &longs;elf might exerci&longs;e its quality of Gravitation, if it really be owner thereof.

SALV. Certainly the Objection of Simplicius is very acute, and therefore its nece&longs;&longs;ary, either that it be unan&longs;werable, or that the Solution be no le&longs;&longs;e acute. That that Air, which compre&longs;­&longs;ed, appeared to weigh as much as that &longs;and, left at liberty in its Element is no longer to weigh any thing as the Sand doth, is a thing manife&longs;t: and therefore for making of &longs;uch an Experiment, its requi&longs;ite to choo&longs;e a place and Medium wherein the Air as well as the Sand might weigh: for, as hath &longs;everal times been &longs;aid, the Medium &longs;ub&longs;tracts from the Weight of every Matter that is im­merged therein, &longs;o much, as &longs;uch another quantity of the &longs;aid Medium, as is that of the ma&longs;&longs;e immer&longs;ed, weigheth: &longs;o that the Air depriveth the Air of all its Gravity. The operation, there­fore, to the end it were made exactly, ought to be tried in a Va­cuum, wherein every grave Body would exerci&longs;e its Moment without any diminution. In ca&longs;e therefore, Simplicius, that we &longs;hould weigh a portion of Air in a Vacuum, would you then be convinced and a&longs;&longs;ured of the bu&longs;ine&longs;&longs;e?

The Air compre&longs;­&longs;ed and violently pent up, weigheth in a Vacuum; and how its weight is to be e&longs;timated.

SIMP. Verily I &longs;hould: but this is to defire, or enjoyn that which is impo&longs;&longs;ible.

SALV. And therefore the obligation mu&longs;t needs be great that you owe to me, when ever I &longs;hall for your &longs;ake effect an impo&longs;&longs;ibi­lity: but I will not &longs;ell you that which I have already given you: for we, in the foregoing Experiment, weigh the Air in a Vacuum,and not in the Air, or in any other Replete Medium. That from the Ma&longs;s, Simplicius, that in the fluid Medium is immerged certain Gravity is &longs;ub&longs;tracted by the &longs;aid Medium, this commeth to pa&longs;s by rea&longs;on that it re&longs;i&longs;teth its being opened, driven back, and in a word commoved; a &longs;ign of which is its pronene&longs;s to return in&longs;tant­ly to fill the Space up again, that the immer&longs;ed ma&longs;s occupied in it, as &longs;oon as ever it departeth thence; for if it &longs;uffered not by that immer&longs;ion, it would not operate again&longs;t the &longs;ame. Now tell me, when you have in the Air the Bottle before filled with the &longs;ame Air naturally contained therein, what divi&longs;ion, repul&longs;e, or, in &longs;hort, what mutation doth the external ambient Air receive from the &longs;e­cond Air that was newly infu&longs;ed with force into the Ve&longs;&longs;el? Doth it enlarge the Bottle, whereupon the Ambient ought the more to retire it &longs;elf to make room for it? Certainly no: And therefore we may &longs;ay, that the &longs;econd Air is not immer&longs;ed in the Ambient, not occupying any Space therein; but is as if it was in a Vacuum,nay more, is really con&longs;tituted in it, and is placed in Vacuities that were not repleted by the former un-conden&longs;ed Air. And, really, I know not how to di&longs;cern any difference between the two Con&longs;ti tutions of Inclo&longs;ed and Ambient, whil&longs;t in this the Ambient doth no-ways pre&longs;s the Inclo&longs;ed, and in that the Inclo&longs;ed doth not re­repul&longs;e the Ambient: and &longs;uch is the placing of any matter in a Vacuum, and the &longs;econd Air compre&longs;sed in the Flask. The weight therefore that is found in that &longs;ame conden&longs;ed Air, is the &longs;ame that it would have, were it freely di&longs;tended in a Vacuum. Tis true in­deed, that the weight of the Sand that weigheth again&longs;t it, as ha­ving been in the open Air, would in a Vacuum have been a little more than ju&longs;t &longs;o heavy; and therefore it is nece&longs;&longs;ary to &longs;ay, that the weighed Air is in reality &longs;omewhat le&longs;&longs;e heavy than the Sand that counterpoi&longs;eth it, that is, &longs;o much, by how much the like quantity of Air would weigh in a Vacuum.

SIMP. I had thought that there was &longs;omething to have been wi&longs;hed for in the Experiments before produced; but now I am thorowly &longs;atisfied.

The difference, though very great, of the Gravity of Moveables hath no part in differer­cing their Veloci­ties.

SALV. The things by me hitherto alledged, and in particular, this, That the difference of Gravity, although exceeding great, hath no part in diver&longs;ifying the Velocities of Moveables, &longs;o that, notwith&longs;tanding any thing depending on that, they would all move with equal Celerity, is &longs;o new, and at the fir&longs;t apprehen&longs;i­on &longs;o remote from probability, that, were there not a way to de­lucidate it, and make it as clear as the Sun, it would be better to pa&longs;&longs;e it over in &longs;ilence, than to divulge it: therefore &longs;eeing that I have let it e&longs;cape from me, its fit that I omit neither Expe­riment nor Rea&longs;on that may corroborate it.

SAGR. Not onely this, but many other al&longs;o of your A&longs;&longs;erti­ons are &longs;o remote from the Opinions and Doctrines commonly received, that &longs;ending them abroad, you would &longs;tir up a great number of Antagoni&longs;ts: in regard, that the innate Di&longs;po&longs;ition of Men doth not &longs;ee with good eyes, when others in their Studies di&longs;cover Truths or Fallacies, that were not di&longs;covered by them­&longs;elves: and with the title of Innovators of Doctrines, little plea­&longs;ing to the ears of many, they &longs;tudy to cut tho&longs;e knots which they cannot untie, and with &longs;ub-terranean Mines to blow up tho&longs;e Structures, which have been with the ordinary Tools by patient Architects erected: but with us here, who are far from any &longs;uch thoughts, your Experiments and Arguments are &longs;ufficient to give full &longs;atisfaction: yet neverthele&longs;&longs;e, if &longs;o be you have other more palpable Experiments, and more convincing Rea&longs;ons we would very gladly hear them.

SALV. The Experiment made with two Moveables, as different in weight as may be, by letting them de&longs;cend from a place on high, thereby to &longs;ee whether their Velocity be equal, meets with &longs;ome difficulty: for if the height &longs;hall be great, the Medium,which is to be opened and laterally repelled by the Impetus of the cadent Body, &longs;hall be of much greater prejudice to the &longs;mall Mo­ment of the light Moveable, than to the violence of the heavy one; whereupon in a long way the light one will be left behind: and in a little altitude it might be doubted whether there were really any difference, or if there were, whether it would be &longs;en&longs;ible. Therefore I have oft been thinking to reiterate the de­&longs;cent &longs;o many times from &longs;mall heights, and to accumulate toge­ther &longs;o many of tho&longs;e minute differences of time, as might inter­cede between the arrival or fall of the heavy Body to the ground, and the arrival of the light one, which &longs;o conjoyned, would make a time not onely ob&longs;ervable, but ob&longs;ervable with much facility Moreover, that I might help my &longs;elf with Motions as &longs;low as po&longs;­&longs;ible may be, in which the Re&longs;i&longs;tance of the Medium operates le&longs;&longs;e in altering the effect that dependeth on &longs;imple Gravity, I have had thoughts to cau&longs;e the Moveable to de&longs;cend upon a de­clining Plane, not much rai&longs;ed above the Plane of the Horizon; for upon this, no le&longs;&longs;e than in perpendicularity, we may di&longs;cover that which is done by Grave Bodies different in weight: and pro­ceeding farther, I have de&longs;ired to free my &longs;elf from any what&longs;o­ever impediment, that might ari&longs;e from the Contact of the &longs;aid Moveables upon the &longs;aid declining Plane: and la&longs;tly, I have ta­ken two Balls, one of Lead, and one of Cork, that above an hun­dred times more grave than this, and have fa&longs;tened them to two &longs;mall threads, each equally four or five yards long, tyed on high: and having removed a&longs;wel the one as the other Ball from the &longs;tate of Perpendicularity, I have let them both go in the &longs;ame Moment, and they de&longs;cending by the Circumferences of Circles de&longs;cribed by the equal Strings their Semidiameters, and having pa&longs;&longs;ed beyond the Perpendicular, they afterwards by the &longs;ame way returned back, and reiterating the&longs;e Vibrations, and re­turns of them&longs;elves neer an hundred times, they have &longs;hewn ve­ry &longs;en&longs;ibly, that the grave Pendulum moveth &longs;o exactly under the time of the light one, that it doth not in an hundred, no nor in a thou&longs;and Vibrations, anticipate the time of one &longs;mall moment, but that they keep an equal pa&longs;&longs;e in their Recur&longs;ions. They al&longs;o &longs;hew the Operation of the Medium, which conferring &longs;ome im­pediment on the Motion, doth much more dimini&longs;h the Vibrati­ons of the Cork, than that of the Lead: not that it maketh them more or le&longs;&longs;e frequent, nay, when the Arches pa&longs;&longs;ed by the Cork were not of above five or &longs;ix degrees, and tho&longs;e of the Lead fif­ty, they did pa&longs;s them under the &longs;ame times.

SIMP. If this be &longs;o, how is it then that the Velocity of the Lead is not greater than that of the Cork? that pa&longs;&longs;ing a jour­ney of &longs;ixty degrees, in the time that this pa&longs;seth hardly &longs;ix?

SALV. But what would you &longs;ay, Simplicius, in ca&longs;e they &longs;hould both di&longs;patch their Recur&longs;ions in the &longs;ame time, when the Cork being removed thirty degrees from the Perpendicular, &longs;hould pa&longs;s an arch of &longs;ixty, and the Lead removed from the &longs;ame middle point onely two degrees, &longs;hould run an arch of four? would not then the Cork be &longs;o much more &longs;wift than the Lead? and yet Experience &longs;hews that &longs;o it happeneth: therefore ob&longs;erve, The Pendulum of Lead being carried v. gr. fifty degrees from the Perpendicular, and thence let go, &longs;wingeth, and pa&longs;&longs;ing beyond the Perpendicular, neer fifty more degrees, de&longs;cribeth an arch of well neer an hundred degrees; and returning of its &longs;elf back again, it de&longs;cribeth another arch, not much le&longs;&longs;e than the former, and continuing its Vibrations, after a great number of them, it finally returneth to Re&longs;t: Each of tho&longs;e Vibrations are made un­der equal times a&longs;wel tho&longs;e of ninety degrees, as tho&longs;e of fifty, twenty, ten, or four; &longs;o that by con&longs;equence, the Velocity of the Moveable doth &longs;ucce&longs;&longs;ively langui&longs;h and abate, in regard, that under equal times it doth &longs;ucce&longs;&longs;ively pa&longs;&longs;e arches continually le&longs;&longs;er and le&longs;&longs;er. The like, yea the &longs;elf &longs;ame effect is performed by the Cork, hanging by a &longs;tring of the like length, &longs;ave that in a le&longs;&longs;e number of Vibracions it returneth to Re&longs;t, as being le&longs;s apt, by means of its Levity, to overcome the ob&longs;tacle of the Air: and yet neverthele&longs;s all the Vibrations, both great and &longs;mall, are made under times equal to one another, and equal al&longs;o to the times of the times of the Vibrations of the Lead. Whereupon it is true, that if whil&longs;t the Lead pa&longs;&longs;eth an arch of fifty degrees, the Cork pa&longs;seth one but of ten, the Cork is then more &longs;low than the Lead: but it will al&longs;o happen on the other &longs;ide, that the Cork pa&longs;seth the arch of fifty degrees, when the Lead pa&longs;seth but that of ten or &longs;ix; and &longs;o in &longs;everal times the Lead &longs;hall be &longs;wifter onewhile, and the Cork another while: but if the &longs;ame Moveables &longs;hall al&longs;o under the &longs;ame equal times, pa&longs;s arches that are equal, one may then very &longs;afely &longs;ay, that their Velocities are equal.

SIMP. This di&longs;cour&longs;e &longs;eems to me concluding, and not con­cluding, and I finde in my thoughts &longs;uch a Confu&longs;ion, ari&longs;ing from the one-while &longs;wift, another-while &longs;low, another-while ex­treme &longs;low motion of both the one and other Moveable; as that it permits me not to di&longs;cern clearly, whether it be true, That their Velocities are alwaies equal.

SAGR. Give me leave, I pray you, Salviatus, to interpo&longs;e two words. And tell me, Simplicius, whether you admit, that it may be &longs;aid with ab&longs;olute verity that the Velocities of the Cork and of the Lead are equal, in ca&longs;e, that both of them departing at the &longs;ame moment from Re&longs;t, and moving by the &longs;ame declivities, they &longs;hould alwaies pa&longs;&longs;e equal Spaces in equal times?

SIMP. This admits of no doubt, nor can it be contradicted.

SAGR. It hapneth now in the Pendulums that each of them pa&longs;&longs;eth now &longs;ixty degrees, now fifty, now thirty, now ten, now eight, four, and two; and when each of them pa&longs;&longs;eth the Arch of &longs;ixty degrees they pa&longs;&longs;e it in the &longs;ame time; in the Arch of fifty the &longs;ame time is &longs;pent by both the one and the other Moveable; &longs;o in the Arch of thirty, of ten, and of the re&longs;t: and therefore it is con­cluded, that the Velocity of the Lead in the Arch of &longs;ixty degrees, is equal to the Velocity of the Cork in the &longs;ame Arch of &longs;ixty de­grees: and that the Velocities in the Arch of fifty, are likewi&longs;e equal to one the other, and &longs;o in the re&longs;t. But it is not &longs;aid, that the Velocity that is exerci&longs;ed in the Arch of &longs;ixty is equal to the Ve­locity that is exerci&longs;ed in the Arch of fifty, nor this to that of the Arch of thirty. But the Velocities are alwaies le&longs;&longs;er, in the le&longs;&longs;er Arches. And this is collected from our &longs;en&longs;ibly &longs;eeing the &longs;ame Moveable con&longs;ume as much time in pa&longs;&longs;ing the great Arch of &longs;ixty degrees, as in pa&longs;&longs;ing the le&longs;&longs;er of fifty, or the lea&longs;t of ten: and, in a word, in their being all pa&longs;&longs;ed alwaies under equal times. It is true therefore, that both the Lead and the Cork &longs;ucce&longs;&longs;ively retard the Motion, according to the Diminution of the Arches, but yet do not alter their harmony in keeping the equality of Velocity in all the &longs;ame Arches by them pa&longs;&longs;ed. I de&longs;ired to &longs;ay thus much, more to try whether I have rightly apprehended the Conceit of Salvia­tus, than out of any nece&longs;&longs;ity that I thought Simplicius to &longs;tand in of a more plain Explanation than that of Salviatus, which is, as in all other things, extreamly clear, and &longs;uch, that, it being fre­quent with him to re&longs;olve Que&longs;tions, in appearance not only ob­&longs;cure, but repugnant to Nature, and to the Truth, with Rea&longs;ons, or Ob&longs;ervations, or Experiments very trite and familiar to every one, it hath (as I have under&longs;tood from divers) given occa&longs;ion to one of the mo&longs;t e&longs;teemed Profe&longs;&longs;ors of our Age to put the le&longs;&longs;e e&longs;teem upon his Novelties, holding them to have as much of Sor­didne&longs;&longs;e, for that they depend on over low and popular Funda­mentals: as if the mo&longs;t admirable and mo&longs;t-to-be-prized Proper­ty of the Demon&longs;trative Sciences, were not to &longs;pring and ari&longs;e from Principles known, under&longs;tood, and granted by every one. But let us, for all that, continue to banquet our &longs;elves with this diet that is &longs;o light of dige&longs;tion; and &longs;uppo&longs;ing that Simplicius is fully &longs;atisfied in under&longs;tanding and admitting, That the intern Gravity of different Moveables hath no &longs;hare in differencing their Veloci­ties, &longs;o that all of them, for ought that dependeth on that, would move with the &longs;ame Velocities; tell us, Salviatus, in what you place the &longs;en&longs;ible and apparent inequalities of Motion; and an­&longs;wer to that In&longs;tance that Simplicius produceth, and which I like­wi&longs;e confirm, I mean, of &longs;eeing a Cannon Bullet move more &longs;wift­ly than a drop of Bird-&longs;hot, for the difference of Velocity &longs;hall be but &longs;mall, in re&longs;pect of that which I object again&longs;t you of Movea­bles of the &longs;ame matter, of which &longs;ome of the greater will de&longs;cend in a Medium, in le&longs;&longs;e than one beat of the Pul&longs;e, that &longs;pace, that others which are le&longs;&longs;er will not pa&longs;&longs;e in an hour, nor in four, nor in twenty; &longs;uch are pebbles and minute gravel-&longs;tones, e&longs;pecially, that &longs;mall &longs;and which muddieth the Water; in which Mediumthey will not de&longs;cend in many hours &longs;o much as two fathoms, which Stones, and tho&longs;e of no great bigne&longs;&longs;e, do pa&longs;&longs;e in one beat of the Pul&longs;e.

SALV. That which the Medium operates, in retarding Movea­bles, the more according as they are compared to one another, le&longs;s grave in &longs;pecie, hath been already declared, &longs;hewing that it pro­ceeds from the &longs;ub&longs;traction of weight. But how one and the &longs;ame Medium can with &longs;o great difference dimini&longs;h the Velocity in Moveables that differ only in Magnitude, although they are of the &longs;ame Matter, and of the &longs;ame Figure, requireth for its expli­cation a more &longs;ubtil di&longs;cour&longs;e, than that which &longs;ufficeth for under­&longs;tanding how the more dilated Figure of the Moveable, or the Motion of the Medium that is made contrary to the Moveable, re­

tardeth the Velocity of the &longs;aid Moveable. I reduce the cau&longs;e of the &longs;aid Problem to the Scabro&longs;ity, and Poro&longs;ity, that is common­ly, and, for the mo&longs;t part, nece&longs;&longs;arily found in the Superficies of Solid Bodies, the which Scabro&longs;ities, in their Motion, go repul&longs;ing and commoving the Air, or other Ambient Medium: of which we have an evident te&longs;timony, in that we hear the Bodies, though made as round as is po&longs;&longs;ible for them to be, to hum whil&longs;t they pa&longs;&longs;e ve­ry &longs;wiftly thorow the Air; and they are not only heard to hum, but to whir and whi&longs;tle, if there be but in them &longs;ome more than ordi­nary cavity or prominency. We &longs;ee al&longs;o, that in turning round every rotund Solid maketh a little wind: And what need more? Do we not hear a notable whirring, and in a very &longs;harp Accent, made by a Top, while it turneth round on the ground with great Celerity? The &longs;hrilne&longs;s of which whizzing groweth flatter accor­ding as the Velocity of the Vertigo doth by degrees more and more &longs;lacken: a nece&longs;&longs;ary Argument likewi&longs;e of the commotion and percu&longs;&longs;ion of the Air by tho&longs;e (though very &longs;mall) Scabro&longs;i­ties of their Superficies. It is not to be doubted, but that the&longs;e in the de&longs;cent of Moveables, grating upon, and repul&longs;ing the fluid Am­bient, procure retardment in the Velocity, and &longs;o much the greater, by how much the Superficies &longs;hall be greater, as is that of le&longs;&longs;er Solids compared to bigger.

The greater or le&longs;s Scabro&longs;ity and Po­ro&longs;ity of the Super­ficies of Movea­bles, a probable cau&longs;e of their grea­ter or le&longs;&longs;er Retar­dation.

SIMP. Stay, I pray you, for here I begin to be at a lo&longs;&longs;e: for though I under&longs;tand and admit, that the Confrication of the Medi­um with the Superficies of the Moveable retardeth the Motion, and that it more retardeth it where (ceteris paribus) the Superficies is greater, yet do I not comprehend upon what ground you call the Superficies of le&longs;&longs;er Solids greater: & farthermore if, as you affirm, the greater Superficies ought to cau&longs;e greater retardment, the greater Solids ought to be the &longs;lower, which is not &longs;o: but this Objection may ea&longs;ily be removed, by &longs;aying, that although the greater hath a greater Superficies, it hath al&longs;o a greater Gravity, upon which the impediment of the greater Superficies hath not &longs;o much more prevalent influence, than the impediment of the le&longs;&longs;er Superficies hath upon the le&longs;&longs;er Gravity, as that the Velocity of the greater Solid &longs;hould become the le&longs;&longs;er. And therefore I &longs;ee no rea&longs;on why one &longs;hould alter the equality of the Velocities, whil&longs;t, that looking how much the Moving Gravity dimini&longs;heth, the faculty of the Re­tarding Superficies doth dimini&longs;h at the &longs;ame rate.

SALV. I will re&longs;olve all that which you object in one word. Therefore, Simplicius, you will without controver&longs;ie admit, that when, of two equal Moveables of the &longs;ame Matter, and alike in Fi­gure (which undoubtedly would move with equal &longs;wiftne&longs;&longs;e) as well the Gravity, as the Superficies of one of them dimini&longs;heth, (yet &longs;till retaining the &longs;imilitude of Figure) the Velocity like­wi&longs;e, for the &longs;ame rea&longs;on, would not be dimini&longs;hed in that which was le&longs;&longs;ened.

SIMP. Really, I think, that it ought &longs;o to follow as you &longs;ay, granting the pre&longs;ent Doctrine with a &longs;alvo &longs;till to our Doctrine, which teacheth, that the greater or le&longs;&longs;er Gravity hath no operati­on in accelerating or retarding Motion.

SALV. And this I confirm; and grant you likewi&longs;e your Po­&longs;ition, from whence, in my opinion, may be inferred, That in ca&longs;e the Gravity dimini&longs;heth more than the Superficies, there may be introduced in the Moveable, in that manner dimini&longs;hed, &longs;ome re­tardment of Motion, and that greater and greater, by how much in proportion, the diminution of the Weight was greater than the di­minution of the Superficies

SIMP. I make not the lea&longs;t que&longs;tion of it.

Solids cannot be dimini&longs;hed at the &longs;ame rate in Super­ficies as in Weight, retaining the &longs;imi­litude of the Fi­gures.

SALV. Now know, Simplicius, that in Solids one cannot di­mini&longs;h the Superficies &longs;o much as the Weight keeping the &longs;imili­tude of Figure. For it being manife&longs;t, that in dimini&longs;hing of grave Solids, the Weight le&longs;&longs;eneth as much as the Bulk, when ever the Bulk happens to be dimini&longs;hed more than the Superficies, (care being had to retain the &longs;imilitude of Figure) the Gravity likewi&longs;e would come to be more dimini&longs;hed than the Superficies. But Geo­metry teacheth us, that there is much greater proportion between the Bulk and the Bulk in like Solids, than between their Superfi­cies. Which for your better under&longs;tanding, I &longs;hall explain in &longs;ome particular ca&longs;e. Therefore fancy to your &longs;elf, for example, a Dye, one of the Sides of which is v. gr. two Inches long, &longs;o that one of its Surfaces &longs;hall be four Square Inches, and all &longs;ix, that is, all its Superficies twenty four Square Inches. Then &longs;uppo&longs;e the &longs;ame Dye at three &longs;awings cut into eight &longs;mall Dice, the Side of every one of which will be one Inch, and one of its Surfaces an Inch Square, and its whole Superficies &longs;ix Square Inches, of which the whole Dye contained twenty four in its Superficial content. Now, you &longs;ee, that the Superficial content of the little Dye is the fourth part of the Superficial content of the great one, (for &longs;ix is the fourth part of twenty four) but the Solid content of the &longs;aid Dye is only the eighth part: therefore the Bulk, and con&longs;equently the Weight, doth much more dimini&longs;h than the Superficies. And if you &longs;ubdivide the little Dye into eight others, we &longs;hall have for the whole Superficial content of one of the&longs;e, one and an half Square Inches, which is the &longs;ixteenth part of the Superficies of the fir&longs;t Dye; but its Bulk, or Ma&longs;s, is only the &longs;ixty fourth part of that. You &longs;ee therefore, how that in only the&longs;e two divi&longs;ions the Bulks decrea&longs;e four times fa&longs;ter than their Superficies: and if we &longs;hould pro&longs;ecute the Subdivi&longs;ion, untill that we had reduced the fir&longs;t So­lid into a &longs;mall powder, we &longs;hould find the Gravity of the minute Atomes to be le&longs;&longs;ened an hundred and an hundred times more than their Superficies. And this which I have exemplified in Cubes, hapneth in all like Solids, the Bulks of which are in Se&longs;­quialter proportion of their Superficies. You &longs;ee, therefore, in how much greater proportion the Impediment of the Contact of the Superficies of the Moveable with the Medium encrea&longs;eth in &longs;mall Moveables, than in greater: and if we &longs;hould add, that the Sca­bro&longs;ities in the very &longs;mall Superficies of the minute Atomes are not happily le&longs;&longs;er than tho&longs;e of the Superficies of greater Solids, that are diligently poli&longs;hed, ob&longs;erve how fluid, and void of all Re­&longs;i&longs;tance being opened, the Medium is required to be, when it is to give pa&longs;&longs;age to &longs;o feeble a Virtue. And therefore take notice, Sim­plicius, that I did not equivocate, when even now I &longs;aid, That the Superficies of le&longs;&longs;er Solids is greater, in compari&longs;on of that of bigger.

SIMP. I am wholly &longs;atisfied: and I verily believe, that if I were to begin my Studies again, I &longs;hould follow the Coun&longs;el of Plato,and enter my &longs;elf fir&longs;t in the Mathematicks, which I &longs;ee to proceed very &longs;crupulou&longs;ly, and refu&longs;e to admit any thing for certain, &longs;ave that which they nece&longs;&longs;arily demon&longs;trate.

SAGR. I have taken great delight in this Di&longs;cour&longs;e; but, be­fore we pa&longs;&longs;e any further, I would be glad to be &longs;atisfied in one particular, which newly came into my thoughts, when but ju&longs;t now you &longs;aid, that Like-Solids are in Se&longs;quialter proportion to their Superficies for I have &longs;een, and under&longs;tood the Propo&longs;ition

with its Demon&longs;tration, in which it is proved, That the Superficies of Like-Solids are in duplicate proportion of their Sides; and ano­ther that proveth the &longs;ame Solids to be in triple proportion of the &longs;ame Sides; but the proportion of Solids to their Superficies, I do not remember that I ever &longs;o much as heard it mentioned.

Solids are to each other in Se&longs;quial­ter proportion to their Superficies.

SALV. You your &longs;elf have an&longs;wered and declared the doubt. For that which is triple of a thing of which another is double, doth it not come to be Se&longs;quialter of this double? Yes doubtle&longs;&longs;e. Now, if Superficies are in double proportion of the Lines, of which the Solids are in triple proportion, may not we &longs;ay, That the Solids are in Se&longs;quialter proportion of their Superficies?

SAGR. I under&longs;tand you very well. And although other par­ticulars, pertaining to the matter of which we have treated, do re­main for me to ask, yet if we &longs;hould thus run from one Digre&longs;&longs;ion to another, it will be late before we &longs;hould come to the Que&longs;tions principally intended, which concern the diver&longs;ities of the Acci­dents of the Re&longs;i&longs;tances of Solids again&longs;t Fraction; and therefore, if you &longs;o plea&longs;e, we may return to the fir&longs;t Theme, which we pro­po&longs;ed in the beginning.

SALV. You &longs;ay very well; but the &longs;o many, and &longs;o different things that have been examined, have &longs;toln &longs;o much of our time, that there is but little of it left in this day to &longs;pend in our other principal Argument, which is full of Geometrical Demon&longs;trati­ons that are to be con&longs;idered with attention: &longs;o that I &longs;hould think it were better to adjourn our meeting till to morrow, as well for this which I have told you, as al&longs;o becau&longs;e I might bring with me &longs;ome Papers, on which I have, in order, &longs;et down the Theorems and Problems, in which are propo&longs;ed and demon&longs;trated the different Pa&longs;&longs;ions of this Subject, which, it may be, would not otherwi&longs;e with requi&longs;ite Method come into my mind.

SAGR. I very gladly comply with your advice, and &longs;o much the more willingly, in regard that, for a Conclu&longs;ion of this daies Con­ference, I &longs;hall have time to hear you re&longs;olve &longs;ome doubts that I find in my mind concerning the Point la&longs;t handled. Of which one is, Whether we are to hold, that the Impediment of the Mediummay be &longs;ufficient to a&longs;&longs;ign bounds to the Acceleration of Bodies of very grave Matter, that are of great Bulk, and of a Spherical Figure: and I in&longs;tance in the Spherical Figure, that I might take that which is contained under the lea&longs;t Superficies, and therefore le&longs;&longs;e &longs;ubject to Retardment. Another &longs;hall be, touching the Vibrations of Pen­dulums, and this hath many heads: One &longs;hall be, Whether all, both Great, Mean, and Little, are made really and preci&longs;ely under equal Times: And another, What is the proportion of the Times of Moveables, &longs;u&longs;pended at unequal &longs;trings, of the Times of their Vibrations I mean.

SALV. The Que&longs;tions are ingenious, and, like as it is incident to all Truths, I &longs;uppo&longs;e, that, which ever of them we handle, it will draw after it &longs;o many other Truths, and curious Con&longs;equences, that I cannot tell whether the remainder of this day may &longs;uffice for the di&longs;cu&longs;&longs;ing of them all.

SAGR. If they &longs;hall be but as delightful as the precedent, it would be more grateful for me to employ as many daies, not to &longs;ay, hours, as it is unto night, and I believe that Simplicius will not be cloy'd with &longs;uch Argumentations as the&longs;e.

SIMP. No certainly: and e&longs;pecially, when the Que&longs;tions trea­ted of are Phy&longs;ical, touching which we read not the Opinions or Di&longs;cour&longs;es of other Philo&longs;ophers.

Any Body, of any Figure, Greatne&longs;s, and Gravity, is checked by the Re­nitence of the Me­dium, though ne­ver &longs;o tenuous, in &longs;uch &longs;ort, that the Motion continuing, it is reduced to equability.

SALV. I come therefore to the fir&longs;t, affirming without any hæ&longs;itation, that there is not a Sphere &longs;o big, nor of Matter &longs;o grave, but that the Renitence of the Medium, though very tenuous, checks its Acceleration, and in the continuation of the Motion reduceth it to Equability, of which we may draw a very clear Argument from Experience it &longs;elf. For if any falling Moveable were able in its continuation of Motion to attain any degree of Velocity, no Velocity that &longs;hould be conferred upon it, could be &longs;o great but that it would depo&longs;e it, and free it &longs;elf of it by help of the Impe­diment of the Medium. And thus, a Cannon-bullet, that had de &longs;cended through the Air, v. gr. four yards, and had, for example, acquired ten degrees of Velocity, and that with the&longs;e &longs;hould enter into the Water, in ca&longs;e the Impediment of the Water were not able to prohibit &longs;uch a certain Impetus in the Ball, it would en­crea&longs;e it, or at lea&longs;t would continue it unto the bottom; which is not ob&longs;erved to en&longs;ue: nay, the Water, although it were but a few fathoms in depth, would impede and debilitate it in &longs;uch a man­ner, that it will make but a &longs;mall impre&longs;&longs;ion in the bottom of the River or Lake. It is therefore manife&longs;t, that that Velocity, of which the Water had ability to deprive it in a very &longs;hort way, would never be permitted to be acquired by it, though in a depth of a thou&longs;and Fathoms. And why &longs;hould it be permitted to gain it in a thou&longs;and, to be taken from it again in four? What need we more? Do we not &longs;ee the immen&longs;e Impetus of the Ball, &longs;hot from the Cannon it &longs;elf, to be in &longs;uch a manner flatted by the interpo­&longs;ition of a few Fathom of Water, that without any harm to the Ship, it but very hardly reacheth to make a dent in it? The Air al­&longs;o, though very yielding, doth neverthele&longs;&longs;e repre&longs;&longs;e the Velocity of the falling Moveable, although it be very heavy, as we may by &longs;uch like Experiments collect; for if from the top of a very high Tower we &longs;hould di&longs;charge a Mu&longs;quet downwards, this will make a le&longs;&longs;er impre&longs;&longs;ion on the ground, than if we &longs;hould di&longs;charge the Mu&longs;quet at the height of four or &longs;ix yards above the Plane: an evident &longs;ign, that the Impetus, wherewith the Bullet i&longs;&longs;ueth from the Gun, di&longs;charged on the top of the Tower, doth gradually di­mini&longs;h in de&longs;cending thorow the Air: therefore the de&longs;cending from any what&longs;oever great height will not &longs;uffice to make it ac­quire that Impetus, of which the Re&longs;i&longs;tance of the Air deprived it, when it had in any manner been conferred upon it. The batte­ry likewi&longs;e that the force of a Bullet, &longs;hot from a Culverin, &longs;hall make in a Wall at the di&longs;tance of twenty Paces, would not, I be­lieve, be &longs;o great, if the Bullet was &longs;hot perpendicularly from any immen&longs;e Altitude. I believe, therefore, that there is a Bound or term belonging to the Acceleration of every Natural Moveable that departs from Re&longs;t, and that the Impediment of the Medium in the end reduceth it to ^{*} Equality, in which it afterwards alwaies

continueth.

* Or Equability.

SAGR. The Experiments are really, in my opinion, much to the purpo&longs;e: nor doth any thing remain, unle&longs;&longs;e the Adver&longs;ary &longs;hould fortifie him&longs;elf, by denying, that they will hold true in great and ponderous Ma&longs;&longs;es, and that a Cannon-bullet coming from the Concave of the Moon, or from the upper Region of the Air, would make a greater percu&longs;&longs;ion than coming from the Cannon.

SALV. There is no que&longs;tion, but that many things may be objected, and that they may not be all &longs;alved by Experiments; ne­verthele&longs;&longs;e in this contradiction, me thinks, there is &longs;omething that may fall under con&longs;ideration; &longs;cilicet, that it is very probable, that the Grave Body, falling from an Altitude, acquireth &longs;o much Impetus, at its arrival to the ground, as would &longs;uffice to return it to that height, as is plainly &longs;een in a Pendulum rea&longs;onable weighty, that being removed fifty or &longs;ixty degrees from the Perpendicular, gaineth that Velocity and Virtue which exactly &longs;ufficeth to force it to the like Recur&longs;ion, that little abated, which is taken from it by the Impediment of the Air. To con&longs;titute, therefore, the Cannon­bullet in &longs;uch an Altitude as may &longs;uffice for the acqui&longs;t of an Impe­tus, as great as that which the Fire giveth it in its i&longs;&longs;uing from the Piece, it would &longs;uffice to &longs;hoot it upwards perpendicularly with the &longs;aid Cannon, and then ob&longs;erving, whether in its fall it maketh an impre&longs;&longs;ion equal to that of the percu&longs;&longs;ion made near at hand in its i&longs;&longs;uing forth; but, indeed, I believe, that it would not be any whit near &longs;o forcible. And therefore I hold that the Velocity, which the Bullet hath near to its going out of the Piece, would be one of tho&longs;e that the Impediment of the Air would never &longs;uffer it to acquire, whil&longs;t it &longs;hould with a natural Motion de&longs;cend, leaving the &longs;tate of Re&longs;t, from any great height. I come now to the other Que&longs;tions belonging to Pendulums, matters which to many would &longs;eem very frivolous, and more e&longs;pecially to tho&longs;e Philo&longs;ophers that are continually bu&longs;ied in the more profound Que&longs;tions of Natural Philo&longs;ophy: yet, notwith&longs;tanding, will not I contemn them, being encouraged by the Example of Ari&longs;totle him&longs;elf, in whom I admire this above all things; that he hath not, as one may &longs;ay, omitted any matter that any waies merited con&longs;ideration, which he hath not &longs;poken of: and now upon the Que&longs;tions you propounded, I think I can tell you a certain conceit of mine upon &longs;ome Problems con­cerning Mu&longs;ick, a noble Subject, of which &longs;o many famous men, and Ari&longs;totle him&longs;elf, have written; and touching it, he con&longs;ide­reth many curious Problems: &longs;o that if I likewi&longs;e &longs;hall from &longs;o fa­miliar and &longs;en&longs;ible Experiments, draw Rea&longs;ons of admirable acci­dents on the Argument of Sounds, I may hope that my di&longs;cour&longs;es will be accepted by you.

A Grave Body, falling from an Altitude, acqui­reth &longs;o much Im­petus at its arri­val to the ground, as in all probabili­ty, would &longs;uffice to recarry it to the &longs;ame height from whence it fell.

SAGR Not only accepted, but by me, in particular, mo&longs;t pa&longs;­&longs;ionately de&longs;ired, in regard that I taking a great delight in all Mu­&longs;ical In&longs;truments, and being rea&longs;onably well in&longs;tructed concerning Con&longs;onances, have alwaies been ignorant and perplexed with endeavouring to know, whence it cometh that one &longs;hould more plea&longs;e and delight me than another; and that &longs;ome not only pro­cure me no delight, but highly di&longs;plea&longs;e me: the trite Ptoblem al­&longs;o of the two Chords &longs;et to an Uni&longs;on, one of which moveth and actually &longs;oundeth at the touching of the other, I al&longs;o am unre&longs;ol­ved in: nor am I very clearly informed concerning the Forms of Con&longs;onances, and other particularities.

SALV. We will &longs;ee, if from the&longs;e our Peudulums one may ga­ther any &longs;atisfaction in all the&longs;e Doubts. And as to the fir&longs;t Que­&longs;tion, that is, Whether the &longs;ame Pendulum doth really and punctu­ally perform all its Vibrations, great, le&longs;&longs;er, and lea&longs;t, under Times preci&longs;ely equal; I refer my &longs;elf to that which I have heretofore learnt from our Academian, who plainly demon&longs;trateth, that the Moveable that &longs;hould de&longs;cend along the Chords, that are Subten­&longs;es to any Arch, would nece&longs;&longs;arily pa&longs;&longs;e them all in equal Times, as well the Subten&longs;e under an hundred and eighty degrees, (that is, the whole Diameter) as the Subten&longs;es of an hundred, &longs;ixty, ten, two, or half a degree, or of four minutes: &longs;till &longs;uppo&longs;ing that they all determine in the lowe&longs;t Point touching the Horizontal Plane. Next as to the de&longs;cendents by the Arches of the &longs;ame Chords eli­vated above the Horizon, and that are not greater than a Qua­drant, that is, than ninety degrees, Experience likewi&longs;e &longs;hews, that they pa&longs;&longs;e all in Times equal, but yet &longs;horter than the Times of the pa&longs;&longs;ages by the Chords: an effect which hath &longs;o much of won­der in it, by how much at the fir&longs;t apprehen&longs;ion one would think the contrary ought to follow: For the terms of the beginning, and the end of the Motion being common, and the Right-Line be­ing the &longs;horte&longs;t, that can be comprehended between the &longs;aid Terms, it &longs;eemeth rea&longs;onable, that the Motion made by it &longs;hould be fini&longs;hed in the &longs;horte&longs;t Time, which yet is not &longs;o: but the &longs;hor­te&longs;t Time, and con&longs;equently, the &longs;wifte&longs;t Motion, is that made by the Arch of which the &longs;aid Right-Line is Chord. In the next place, as to the Times of the Vibrations of Moveables, &longs;u&longs;pended by &longs;trings of different lengths, tho&longs;e Times are in Subduple pro­portion to the lengths of the &longs;trings, or, if you will, the lengths are in duplicate proportion to the Times, that is, are as the Squares of the Times: &longs;o that if, for example, the Time of a Vibration of one Pendulum is double to the Time of a Vibration of another, it followeth, that the length of the &longs;tring of that is quadruple to the length of the &longs;tring of this. And in the Time of one Vibration of that, another &longs;hall then make three Vibrations, when the &longs;tring of that &longs;hall be nine times as long as the other. From whence doth follow, that the length of the &longs;trings have to each other the &longs;ame proportion, that the Squares of the Numbers of the Vibrations that are made in the &longs;ame Times have.

Moveables de&longs;cen­ding along the Chords, that are Subten&longs;es to any Arch of a Circle, pa&longs;&longs;e as well the greater as the le&longs;­&longs;er Chords in equal Times.

Moveables andPendula de&longs;cend­ing along the Ar­ches of the &longs;ame Chords, elivated as far as 90 deg. pa&longs;s the &longs;aid Arches in Times equal, but that are &longs;horter than the tran&longs;iti­ons along the Chords.

The Times of the Vibrations of Mo­vables, hanging at alonger or &longs;horter thread, are to one another in propor­tion &longs;ubduple the lengths of the &longs;trings, at which they hang.

SAGR. Then, if I have rightly under&longs;tood you, I may ea&longs;ily know the length of a &longs;tring, hanging at any never-&longs;o-great height, although the &longs;ublime term of the &longs;u&longs;pen&longs;ion were invi&longs;ible to me, and I only &longs;aw the other lower extream. For if I &longs;hall fa&longs;ten a weight of &longs;ufficient Gravity to the &longs;aid &longs;tring here below, and &longs;et it on vibrating to and again, and a friend telling &longs;ome of its Recur­&longs;ions, and I at the &longs;ame time tell the Recur&longs;ions of another Movea­ble, &longs;u&longs;pended at a &longs;tring that is preci&longs;ely a yard long, by the Numbers of the Vibrations of the&longs;e Pendula, made in the &longs;ame Time, I will find the length of the &longs;tring. As for example, &longs;uppo&longs;e that in the time that my friend hath counted twenty Recur&longs;ions of the long &longs;tring, I had told two hundred and forty of my &longs;tring, that is one yard long: &longs;quaring the two numbers twenty and two hundred and forty, which are 400, and 57600, I will &longs;ay, that the long &longs;tring containeth 57600 of tho&longs;e Mea&longs;ures, of which my &longs;tring containeth 400. and becau&longs;e the &longs;tring is one &longs;ole yard, I will divide 57600 by 400, and the quotient will be 144, and I will af­firm that &longs;tring to be 144 yards long.

To find the Length of any Rope, or &longs;tring, at which a Moveable hang­eth, by the frequen­cy of its Vibrations

SALV. Nor will you be mi&longs;taken one Inch; and e&longs;pecially, if you take a great Number of Vibrations.

SAGR. You give me frequent occa&longs;ion to admire the Riches, and withal the extraordinary bounty of Nature, whil'&longs;t by things &longs;o common, and, I might in a certain &longs;ence &longs;ay, vile, you go col­lecting of Notions very curious, new, and oftentimes, remote from all imagination. I have an hundred times con&longs;idered the Vi­brations, in particular, of the Lamps in &longs;ome Churches, hanging by very long ropes, when they have been unawares &longs;tirred by any one: but the mo&longs;t that I inferred from that &longs;ame Ob&longs;ervati­on, was the improbability of the Opinion of tho&longs;e who hold, that &longs;uch-like Motions are maintained and continued by the Medi­um, that is by the Air: for it &longs;hould &longs;eem to me, that the Air had a great judgment, and withal but little bu&longs;ine&longs;&longs;e to &longs;pend &longs;o ma­ny hours time in vibrating an hanging Weight with &longs;o much Regu­larity: but that I &longs;hould have learnt, that that &longs;ame Moveable, &longs;u&longs;pended at a &longs;tring of an hundred yards long, being removed from Perpendicularity one while ninety degrees, and another while one degree onely, or half a degree, &longs;hould &longs;pend as much time in pa&longs;&longs;ing this little, as in pa&longs;&longs;ing that great Arch, certainly would never have come into my head, for I &longs;till think, that it bordereth upon Impo&longs;sibility. Now I am in expectation to hear that the&longs;e petty Notions will a&longs;sign me &longs;uch Rea&longs;ons of tho&longs;e Mu&longs;ical Pro­blems, as may, in part at lea&longs;t, give me &longs;atisfaction.

Every Pendulum hath the Time of its Vibration &longs;o li­mited; that it is not po&longs;&longs;ible to make it move under any other Period.

SALV. Above all things, you are to know, that every Pendu­lum hath the Time of its Vibrations &longs;o limited, and prefixed, that it is impo&longs;&longs;ible to make it move under any other Period, than that onely one, which is natural unto it. Let any one take the &longs;tring in hand, to which the Weight is fa&longs;tened, and trie all the wayes he can to encrea&longs;e or decrea&longs;e the frequency of its Vibrations, and he &longs;hall finde it labour in vain: but we may, on the contrary, on a Pendulum, though grave and at re&longs;t, by onely blowing up­on it, conferre a Motion, and a Motion con&longs;iderably great, by reiterating the bla&longs;ts, but under the Time that is properly be­longing to its Vibrations: for if at the fir&longs;t bla&longs;t we &longs;hould have re­moved it from Perpendicularity half an Inch, adding a &longs;econd, after that it being returned towards us, is ready to begin the &longs;e­cond Vibration, we &longs;hould conferre new Motion on it, and &longs;o &longs;ucce&longs;&longs;ively with other bla&longs;ts, but given in Time, and not when the Pendulum is comming towards us (for &longs;o we &longs;hould impede; and not help the Motion) and &longs;o continuing with many Impul­&longs;es, we &longs;hould confer upon it &longs;uch an Impetus, that a greater force by much than that of a bla&longs;t of our breath, will be required to &longs;tay it.

SAGR. I have, from my childhood, ob&longs;erved, that one man a­lone, by means of the&longs;e Impul&longs;es, given in Time, hath been able to towl a very great Bell, and when it was to cea&longs;e, I have &longs;een four or &longs;ix men more lay hold on the Bell-rope, and they have all been rai&longs;ed from the ground: &longs;o many together being unable to arre&longs;t that Impetus, which one alone, with regular Pulls, had con­ferred upon the Bell.

SALV. An example, that declareth my meaning with no le&longs;&longs;e

propriety than this that I have premi&longs;ed, doth &longs;ute to render the rea&longs;on of the admirable Problem of the Chord of the Lute or Viol, which moveth, and maketh not onely that really to &longs;ound, which is tuned to the Uni&longs;on, but that al&longs;o which is &longs;et to an Eighth and a Fifth. The Chord being toucht, its Vibrations begin, and continue all the Time that its Sound is heard to endure: the&longs;e Vibrations make the Air neer adjacent to vibrate and tremble, who&longs;e tremblings and quaverings di&longs;tend them&longs;elves a great way, and &longs;trike upon all the Chords of the In&longs;trument, and al&longs;o of o­thers neer unto it: the Chord that is &longs;et to an Uni&longs;on, with that which is toucht, being di&longs;po&longs;ed to make its Vibrations ^{*} in the &longs;ame Time, beginneth at the fir&longs;t impul&longs;e to move a little, and a &longs;econd, a third, a twentieth, and many more, overtaking it, all in ju&longs;t and Periodick Times, it receiveth at la&longs;t, the &longs;ame Tre­mulation, with that fir&longs;t touched, and one may clearly &longs;ee it go, dilating its Vibrations exactly according to the Pace of its Mo­ver. This Undulation that di&longs;tendeth it &longs;elf thorow the Air, mo­veth, and makes to vibrate, not onely the Chords, but likewi&longs;e any other Body di&longs;po&longs;ed to trembling, and to vibrate in the very Time of the trembling Chord: &longs;o that if we fix in the Sides of the In&longs;trument &longs;everal &longs;mall pieces of Bri&longs;tles, or of other flexible matters, you &longs;hall &longs;ee upon the &longs;ounding of the Viol, now one, now another of tho&longs;e Corpu&longs;cles tremble, according as that Chord is toucht, who&longs;e Vibrations return in the &longs;ame Time: the others will not move at the &longs;triking of this Chord, nor will that Bri&longs;tle tremble at the &longs;triking of another Chord. If with the Bow one &longs;martly &longs;trike the Ba&longs;e-Chord of a Viol, and &longs;et a drinking Gla&longs;&longs;e, thin and &longs;mooth, neer unto it, if the Tone of the Chord be an Uni&longs;on to the Tone of the Gla&longs;&longs;e, the Gla&longs;&longs;e &longs;hall dance, and &longs;en&longs;ibly re-&longs;ound. Again, the ample dilating of the Tremor or Undulation of the Medium about the Body re&longs;ounding, is ap­parently &longs;een in making the Gla&longs;&longs;e to &longs;ound, by putting a little Water in it, and then chafing the brim or edge of it with the tip of the finger: for the included Water is ob&longs;erved to undulate in a mo&longs;t regular order: and the &longs;ame effect will be yet more clearly &longs;een, by &longs;etting the foot of the Gla&longs;&longs;e in the bottom of a rea&longs;o­nable large Ve&longs;&longs;el, in which there is Water as high almo&longs;t as to the brim of the Gla&longs;&longs;e, for making it to &longs;ound, as before, with the Confrication of the finger, we &longs;hall &longs;ee the trembling of the Water to diffu&longs;e it &longs;elf mo&longs;t regularly, and with great Velocity, to a great di&longs;tance round about the Gla&longs;&longs;e; and it hath many times been my fortune, in making a rea&longs;onable big Gla&longs;&longs;e, almo&longs;t full of Water, to &longs;ound as afore&longs;aid, to &longs;ee the Waves in the Water, at fir&longs;t formed with an exact equality; and it hapning &longs;ometimes, that the Tone of the Gla&longs;&longs;e ri&longs;eth an Eighth higher, at the &longs;ame in&longs;tant, I have &longs;een every one of the &longs;aid Waves to divide them&longs;elves in two: an accident that very clearly proveth the forme of the Octave to be the double.

The Chord of a Mu&longs;ical In&longs;tru­ment touched, mo­veth, and maketh the Chords &longs;et to an Uni&longs;on, Fifth and Eighth, with it to &longs;ound; and why.

Sundry Problems touching Mu&longs;ical Proportions, and their Solutions.

* Or under.

SAGR. The &longs;ame hath al&longs;o befaln me more than once, to my delight, and al&longs;o benefit: for I &longs;tood a long time perplexed a­bout the&longs;e Forms of Con&longs;onants, not conceiving, that the Rea­&longs;on, commonly given thereof by the Authours that have hither­to written learnedly of Mu&longs;ick, were &longs;ufficiently convincing, they tell us, that the Diapa&longs;on, that is the Eighth, is contained by the double, the Diapente, which we call the Fifth, by the Se&longs;quialter: for a Chord being di&longs;tended on the ^{*} Monochord, &longs;triking it all; and afterwards &longs;triking but the half of it, by pla­cing a Bridge in the middle, one heareth an Eighth; and if the Bridge be placed at a third of the whole Chord, touching the whole, and then the two thirds, it &longs;oundeth a Fifth; whereupon they infer, that the Eighth is contained between two and one, and the Fifth between three and two. This Rea&longs;on, I &longs;ay, &longs;eemed to me not nece&longs;&longs;arily concluding for the a&longs;&longs;igning ju&longs;tly the double and the Se&longs;quialter, for the natural Forms of the Diapa&longs;on and the Diapente. And that which moved me &longs;o to think, was this. There are three ways, by which we may &longs;harpen the Tone of a Chord: one is, by making it &longs;horter, the other is by di&longs;tending; or making it more ten&longs;e; and the third is by making it thinner. If, retaining the &longs;ame Tention and thickne&longs;&longs;e, we would hear an Eighth, it is nece&longs;&longs;ary to &longs;horten it to one half, which is done by &longs;triking it all, and then half. But if, retaining the &longs;ame length and thickne&longs;&longs;e, we would have it ri&longs;e to an Eighth, by &longs;crewing it higher, it will not &longs;uffice to &longs;tretch it double as much, but we &longs;hall need the quadruple, &longs;o that, if before it was &longs;tretched by a Weight of one pound, it will be needful to fa&longs;ten four pound to it to &longs;harpen it to an Eighth. And la&longs;tly, if, keeping the &longs;ame length and Tention, we would have a Chord, that by being &longs;mal­ler, rendereth an Eighth, it will be nece&longs;&longs;ary, that it retain onely a fourth part of the thickne&longs;&longs;e of the other more Grave. And this which I &longs;peak of the Eighth, that is, that its form taken from the Tention, or from the thickne&longs;&longs;e of the Chord, is in duplicate proportion to that which it receiveth from the length, is to be under&longs;toood of all other Mu&longs;ical Intervals: for that which the length giveth us in a Se&longs;quialter proportion, i. e. by &longs;triking it all, and then the two thirds, if you would have it proceed from the Tention, or from the di&longs;gro&longs;&longs;ing, you mu&longs;t double the Se&longs;qui­alter proportion, taking the double Se&longs;quiquartan: and if the Grave Chord were &longs;tretched by four pound weight, fa&longs;ten to the Acute not &longs;ix, but nine: and, as to the thickne&longs;&longs;e, make the Grave Chord thicker than the Acute, according to the proportion of nine to four, to have the Fifth. The&longs;e being mo&longs;t exact Experi­ments, I thought, that I &longs;aw no rea&longs;on, why the&longs;e Sage Philo&longs;o­phers &longs;hould e&longs;tabli&longs;h the form of the Eighth to be rather the dou­ble, than quadruple; and the Form of the Fifth to be rather the Se&longs;quialter, than the double Se&longs;quiquartan. But becau&longs;e the numbring of the Vibrations of a Chord, which in giving a &longs;ound, are extreme frequent, is altogether impo&longs;&longs;ible, I &longs;hould always have been in doubt, whether or no it were true, that the more Acute Chord of the Eighth, made in the &longs;ame time, double the number of the Vibrations of the more Grave, if the Waves, which may be continued as long as you plea&longs;e, by making the Gla&longs;s to &longs;ound and vibrate, had not &longs;en&longs;ibly &longs;hewn me, that in the &longs;elf &longs;ame moment that (&longs;ometimes) the Sound is heard to ri&longs;e to an Eighth, there are &longs;een to ari&longs;e other Waves more minute, which with infinite &longs;moothne&longs;s cut in the middle each of tho&longs;e fir&longs;t.

* An In&longs;trument of but one &longs;tring; called by Mar­&longs;ennus la Tromper­te Marine.

SALV. An excellent Ob&longs;ervation for di&longs;tingui&longs;hing one by one the Undulations ari&longs;ing from the Tremulation of the re­&longs;ounding Body: which are tho&longs;e that diffu&longs;ing them&longs;elves tho­row the Air, make the titillation upon the Drum of our Ear, that in our Soul becommeth a Sound: But whereas beholding and ob­&longs;erving them in the Water, endure no longer than the confrica­tion of the finger la&longs;teth, and al&longs;o in that time they are not per­manent, but are continually made and di&longs;&longs;olved, would it not be an ingenious undertaking, if one could make, with much exqui&longs;itene&longs;&longs;e, &longs;uch, as would continue a long time; I mean Moneths and Years, &longs;o as to give a man opportunity mea&longs;ure, and with ea&longs;e to number them?

SAGR. I a&longs;&longs;ure you I &longs;hould highly value &longs;uch an Invention.

SALV. The di&longs;covery was accidental, and the Ob&longs;ervation and applicative improvement of it onely were mine, and I hold it to be a Circum&longs;tance of noble Contemplation, althongh a bu&longs;i­ne&longs;&longs;e in its &longs;elf &longs;ufficiently homely. Scraping a Bra&longs;&longs;e Plate with an Iron Chizzel to fetch out &longs;ome Spots, in moving the Chizzel to and again upon it pretty quick, I heard it (once or twice among&longs;t many gratings) to Sibilate and &longs;end forth a whi&longs;tling noi&longs;e, very &longs;hrill and audible: and looking upon the Plate, I &longs;aw a long row of &longs;mall &longs;treaks, parallel to one another, and di&longs;tant from one another by mo&longs;t equal Intervals: returning to my &longs;craping again, I perceived by &longs;everal trials, that in tho&longs;e &longs;crapings, and tho&longs;e onely that whi&longs;tled, the Chizzel left the &longs;treaks upon the Plate: but when the Scraping pa&longs;&longs;ed without any Sibilation, there was not &longs;o much as the lea&longs;t &longs;ign of any &longs;uch &longs;treaks. Re­peating the Experiment &longs;everal times afterwards, &longs;craping now with greater, now with le&longs;&longs;e velocity, the Sibilation hapned to be of a Tone &longs;ometimes acuter, &longs;ometimes graver; and I ob&longs;erved the marks made in the more acute &longs;ounds to be clo&longs;er together, and tho&longs;e of the more grave farther a&longs;under: and &longs;ometimes al&longs;o, according as the &longs;elf &longs;ame &longs;crape was made towards the end, with greater velocity than at the beginning, the &longs;ound was heard to grow &longs;harper, and the &longs;treaks were ob&longs;erved to &longs;tand thicker, but ever with extream neatne&longs;&longs;e, and marked with exact equidi­&longs;tance: and farther-more, in the Sibilating &longs;crapes; I felt the Chizzel to &longs;hake or tremulate in my hand, and a certain chilne&longs;&longs;e to run along my arm; and in &longs;hort, I &longs;aw the &longs;ame effected upon the Toole, which we u&longs;e to ob&longs;erve in whi&longs;pering, and after­wards &longs;peaking aloud, for &longs;ending forth the breath without forming a &longs;ound, we do not perceive any moving in the throat and mouth, in compari&longs;on of that which we di&longs;cern to be in the Wind-pipe and Throat of every one, in &longs;ending forth the voice; and e&longs;pecially in grave and loud Tones. I have likewi&longs;e &longs;ome­times among&longs;t the Chords of the Viols, ob&longs;erved two that were Uni&longs;ons to the Sibilations made by &longs;craping after the manner I told you, and that were mo&longs;t different in Tone, from which two they preci&longs;ely were di&longs;tant a perfect Fifth, and then mea&longs;uring the intervals of the &longs;treaks of both the Scrapes, I &longs;aw the di­&longs;tance that conteined forty five &longs;paces of the one, conteined thirty of the other: which, indeed, is the Form attributed to the Diapente. But here, before I proceed any farther, I will tell you, that of the three manners of rendring a Sound Acute, that which you refer to the &longs;lenderne&longs;&longs;e or finene&longs;&longs;e of the Chord, may with more truth be a&longs;cribed to the Weight. For the alteration ta­ken from the thickne&longs;&longs;e, an&longs;wereth, when the Chords are of the &longs;ame matter; and &longs;o a Gut-&longs;tring to make an Eighth, ought to be four times thicker than the other Gut-&longs;tring; and one of Wier four times thicker than another of Wier. But if I would make an Eighth with one of Wier to one of Gut-&longs;tring, I am not to make it four times thicker, but four times graver, &longs;o that, as to thickne&longs;&longs;e, this of Wier &longs;hall not be four times thicker, but quadruple in Gravity, for &longs;ome times it &longs;hall be more &longs;mall than its re&longs;pon­dent to the Acuter Eighth, that is of Gut-&longs;tring. Hence it com­meth to pa&longs;&longs;e that, &longs;tringing an In&longs;trument with Chords of Gold, and another with Chords of Bra&longs;&longs;e, if they &longs;hall be of the &longs;ame length, thickne&longs;&longs;e, and Tention, Gold being almo&longs;t twice as heavy, the Strings &longs;hall prove about a Fifth more Grave. And here it is to be noted, that the Gravity of the Moveable more re­&longs;i&longs;teth the Velocity, than the thickne&longs;&longs;e doth; contrary to what others at the fir&longs;t would think: for indeed, in appearance, its more rea&longs;onable, that the Velocity &longs;hould be retarded by the Re&longs;i&longs;tance of the Medium again&longs;t Opening in a Moveable thick and light, than in one grave and &longs;lender: and yet in this ca&longs;e it happeneth quite contrary. But pur&longs;uing our fir&longs;t Intent, I &longs;ay, That the ncere&longs;t and immediate rea&longs;ons of the Forms of Mu&longs;ical Intervals, is neither the length of the Chord, nor the Tention, nor the thickne&longs;&longs;e, but the proportion of the numbers of the Vibrations, and Percu&longs;&longs;ions of the Undulations of the Air that beat upon the Drum of our Ear, which it &longs;elf al&longs;o doth tremulate under the &longs;ame mea&longs;ures of Time. Having e&longs;tabli&longs;hed this Point, we may, perhaps, a&longs;&longs;ign a very apt rea&longs;on, whence it commeth, that of tho&longs;e Sounds that are different in Tone, &longs;ome Couples are re­ceived with great delight by our Sence, others with le&longs;s, and others occa&longs;ion in us a very great di&longs;turbance; which is to &longs;eek a rea&longs;on of the Con&longs;onances more or le&longs;&longs;e perfect, and of Di&longs;lo­nances. The mole&longs;tation and har&longs;hne&longs;&longs;e of the&longs;e proceeds, as I believe, from the di&longs;cordant Pul&longs;ations of two different Tones, which di&longs;proportionally &longs;trike the Drum of our Ear: and the Di&longs;&longs;onances &longs;hall be extreme har&longs;h, in ca&longs;e the Times of the Vi­brations were incommen&longs;urable. For one of which take that, when of two Chords &longs;et to an Uni&longs;on, one is &longs;ounded, and &longs;uch a part of another, as is the Side of the Square of its Diameter; a Di&longs;&longs;onance like to the ^{*} Tritone, or Semi-diapente. Con&longs;onan­ces, and with plea&longs;ure received, &longs;hall tho&longs;e Couples of Sounds be, that &longs;hall &longs;trike in &longs;ome order upon the Drum; which order requireth, fir&longs;t, that the Pul&longs;ations made in the &longs;ame Time be commen&longs;urable in number, to the end, the Cartillage of the Drum, may not &longs;tand in the perpetual Torment of a double inflection of allowing and obeying the ever di&longs;agreeing Percu&longs;&longs;ions. Therefore the fir&longs;t and mo&longs;t grateful Con&longs;onance &longs;hall be the Eighth, being, that for every &longs;troke, that the Grave-&longs;tring or Chord giveth upon the Drum, the Acute giveth, two; &longs;o that both beat together in every &longs;econd Vibration of the Acute Chord; and &longs;o of the whole number of &longs;trokes, the one half accord to &longs;trike together, but the &longs;trokes of the Chords that are Uni&longs;ons, alwayes joyn both together, and therefore they are, as if they were of the &longs;ame Chord, nor make they a Con&longs;onance. The Fifth delighteth likewi&longs;e, in regard, that for every two &longs;troaks of the Grave Chord, the Acute giveth three: from whence it followeth, that numbering the Vibrations of the Acute Chord, the third part of that number will agree to beat together; that is, two Solitary ones interpo&longs;e between every couple of Con&longs;onances; and in the Di­ate&longs;&longs;eron there interpo&longs;e three. In the &longs;econd, that is in the Se&longs;-quioctave Tone for every nine Pul&longs;ations, one onely &longs;trikes in Con­&longs;ort with the other of the Graver Chord; all the re&longs;t are Di&longs;cords, and received upon the Drum with regret, and are judged Di&longs;&longs;o­nances by the Ear.

* Or a fal&longs;e Fifth.

SIMP. I could wi&longs;h this Di&longs;cour&longs;e were a little explained.

SALV. Suppo&longs;e this line A B the Space, and dilating of a Vi­bration of the Grave Chord; and the line C D that of the Acute Chord, which with the other giveth the Eighth: and let A B be divided in the mid&longs;t in E. It is manife&longs;t, that the Chords begin­ing to move at the terms A and C, by that time the Acute Vibra­tion &longs;hall be come to the term D, the other

SAGR. I can hold no longer, but mu&longs;t needs expre&longs;&longs;e the con­tent I take in hearing rea&longs;ons &longs;o appo&longs;itely a&longs;&longs;igned of effects that have &longs;o long time held me in darkne&longs;&longs;e and blindne&longs;&longs;e. Now I know why the Uni&longs;on differeth not at all from a &longs;ingle Tone: I &longs;ee why the Eighth is the principal Con&longs;onance, but withal &longs;o like to an Uni&longs;on, that, as an Uni&longs;on, it is taken and cojoyned with others: it re&longs;embleth an Uni&longs;on, for that whereas the Pul­&longs;ations of Chords &longs;et to an Uni&longs;on, keep time in &longs;triking, the&longs;e of the Grave Chord in an Eighth alwayes keep time with tho&longs;e of the Acute, and of the&longs;e one interpo&longs;eth alone, and in equal di&longs;tances, and as, one may &longs;ay, without any variety, whereupon that Con&longs;onance is over &longs;weet. But the Fifth, with tho&longs;e its Counter-times, and with the interpo&longs;ures of two &longs;olitary Pul&longs;a­tions of the Acute Chord, and one of the Grave Chord, between the Couples of Di&longs;cordant Pul&longs;ations, and tho&longs;e three &longs;olitary ones, with an interval of time, as great as the half of that which interpo&longs;eth between each Couple, and the &longs;olitary Percu&longs;&longs;ions of the Acute Chord, maketh &longs;uch a Titillation and Tickling upon the Cartillage of the Drum of the Ear, that al­laying the Dulcity with a mixture of Acrimony, it &longs;eemeth at one and the &longs;ame time to ki&longs;&longs;e and bite.

SALV. It is convenient, in regard I &longs;ee, that you take &longs;uch de­light in the&longs;e Novelties, that I &longs;hew you the way whereby the Eye al&longs;o, and not the Ear alone, may recreate it &longs;elf in beholding the &longs;ame &longs;ports that the Ear feeleth. Su&longs;pend Balls of Lead or o­ther heavy matter on three &longs;trings of different lengths, but in &longs;uch proportion, that while the longer maketh two Vibrations, the &longs;horter may make four, and the middle one three; which will happen, when the longe&longs;t containeth &longs;ixteen feet, or other mea&longs;ures, of which the middle one containeth nine, and the &longs;horte&longs;t four: and removing them all together from Perpendi­cularity, and then letting them go, you &longs;hall &longs;ee a plea&longs;ing In­termixtion of the &longs;aid Pendulums with various encounters, but &longs;uch, that, at every fourth Vibration of the longe&longs;t, all the three will concurre in one and the &longs;ame term together, and then again will depart from it, reiterating anew the &longs;ame Period: the which commixture of Vibrations, is the &longs;ame, that being made by the Chords, pre&longs;ents to the Ear an Eighth, with a Fifth in the mid&longs;t. And if you qualifie the length of other &longs;trings in the like di&longs;po­&longs;ure, &longs;o that their Vibrations an&longs;wer to tho&longs;e of other Mu&longs;ical, but Con&longs;onant Intervals, you &longs;hall &longs;ee other and other Inter­weavings, and alwaies &longs;uch, that in determinate times, and after determinate numbers of Vibrations, all the &longs;trings (be they three, or be they four) will agree to joyn in the &longs;ame moment, in the term of their Recur&longs;ions, and from thence to begin &longs;uch another Period: but if the Vibrations of two or more &longs;trings are either Incommen&longs;urable, &longs;o, that they never return harmoniou&longs;ly to ter­minate determinate numbers of Vibrations, or though they be not Incommen&longs;urable, yet if they return not till after a long time, and after a great number of Vibrations, then the &longs;ight is con­founded in the di&longs;orderly order of irregular Intermixtures, and the Ear with wearine&longs;&longs;e and regret receiveth the intemperate Im­pul&longs;es of the Airs Tremulations, that without Order or Rule, &longs;ucce&longs;&longs;ively beat upon its Drum.

But whither, my Ma&longs;ters, have we been tran&longs;ported for &longs;o many hours by various Problems, and unlook't for Di&longs;cour&longs;es? We have made it Night, and yet we have handled few or none of the points propounded; nay we have &longs;o lo&longs;t our way, that I &longs;car&longs;e remember our fir&longs;t entrance, and that &longs;mall Introduction, which we laid down, as the Hypothe&longs;is and beginning of the fu­ture Demon&longs;trations.

SAGR It will be convenient, therefore, that we break up our Conference for this time, giving our Minds leave to compo&longs;e them&longs;elves in the Nights Repo&longs;e, that we may to Morrow (if you plea&longs;e &longs;o far to favour us) rea&longs;&longs;ume the Di&longs;cour&longs;es de&longs;ired, and chiefly intended.

SALV. I &longs;hall not fail to be here to Morrow at the u&longs;ual hour, to &longs;erve and enjoy you.

The End of the Fir&longs;t Dialogue.