College Notebook

Of the extraction of Pure Square Cubick. Square-square & square-cubick rootes &c

Let the number whose roote is to bee extracted bee pointed makeing the first point under the {unite} & comprizeing soe many numbers under each point as the number hath dimensions as if the number be square-cube tis thus pointed

Then out of the figures of the first point next the left hand extract the greatest roote proper to the power of the number & set that downe in the Quotient which is the first side & is called A. (as the roote quintuplicate of is , & quintuplicate is ) then takeing that roote duely multiplied out of the number (as out of ) with the rest of the numbers to the next point. seeke the seacond side which is found by divideing that number by another number made out of the first side (which is called the Divisor) & this second side I name E. (thus by divideing by after such a maner that may be conteined in the number the product of that division shall be E =

The extraction of the square roote

The extraction of the cube roote

The extraction of the square square roote

The extraction of the Square-Cube roote

Note that the 3^d 4^th 5^th & other figures are found by the same manner that the seacond figure is found onely makeing all the figures found to stand for A the first side & the figure sought for e or the 2^d side

And if roote is found inexpressible in whole numbers then adding ciphers & pointing them from the unite towards the right {kind} as was before explained & soe hold on the works in decimalls.

As for the Divisors they are easily found by the 2^d Table of Powers from a Binomial roote.

If the Number bee of 6.7.8.9.10 &c dimensions The roote may be extracted after the same manner

Of the extraction of Rootes in Affected powers.

The manner of the extraction of rootes in pure & affected powers is verry much alike, especially when the affected powers are decently prepared, that is, when theire affections are not over large & those altogether either affirmative or negative, & the power affirmative, affirmations & negations so mixt that there be noe ambiguity & all fractions & Asymmetry taken away

All the figures in the coefficients & affected power are to be pointed (after the manner before explained in the Analisis of pure powers) according to the degree of theire dimensions the worke onely differs that in pure powers in that the coefficients enter into the divisors

Let the first side be called A.the 2^d be called E. the Roote of the equation {L} the coefficients &c the Power &c & the Operation follows

The analysis of Cubick Equations

The equation supposed .

But the coëfficient maybe greater than the Power soe that it cannot be substracted from it which argues that the cube more propperly affects than is affected. In this case the coëfficient must descend towards the unite soe many points untill it may be substracted, & soe many points as the coëfficient is devolved soe many pricks must be blotted out towards the left hand in the power affected. As the example shows .

Since is greater than make a devolution thus.

To place the unite of the coefficient in its right place in respect of the power make so many pricks above as there are under the power begining at the unit, & if the coefficient be one dimension lesse than the power make a prick on every figure if 2 dimensions les than every other figure if 3 dimensions lesse make it one each third figure &c

If there be many coefficients in the equation each must be placed according to this rule.

Sometimes the coefficient is under a negative sine as & the Analysis is as follows

But sometimes the square coëfficient hath more paires of figures than the cube to be analysed, hath & then there is præfixing so many ciphers to the cube as figures are wanting, the first side will not much differ from the square roote of the coefficient. as

Sometimes though there be as many 2 figures in the coefficient as 3 figures in the cube affected yet the coëfficient may be so greate as to deceive an unwary Analist as in this . where the roote of is which cubed is which added to makes then whose roote the number immediately greater is which is the first side .

But if the coefficient had beene affirmative, then not the aggregate of the facts but the difference must be taken as in this. .

Since the roote of is . which cubed is . & . the roote of which is . The like is observable in equations of higher powers

If the cube be affected with a negative sine as . Then the equation is expressible of 2 rootes: whereof the square of one is <6r> lesse & the square of the other is greater then . & therefore one roote is lesse the other greater then . & in this equation are two rootes whereof one is greater the other lesse then .

Suppose in the former cubick equation the lesse roote be 12. then . or else . & . where is the greater roote.

And in the latter equation if the greater roote be 27. & , c. or . . If there be 4 cubes continually proportionall whose greate extreame is . & the aggregate of the 3 rest is 8072 & the lesse extreame, therefore . the roote of which .is 8 the other roote of the equation

Or having one roote of an equation the Equation may be lessened by division thus or . & one roote is 12. therefore divide this equation by & the Quote is an equation conteining the other roote viz: .

Propositiones Geometricæ. Franc: Vietæ.

prop 1

prop 2

& if : then:

prop 3. If . then

prop 3. To find two meane proportionalls {twixt} & . On the center with the radius describe the circle . inscribe . Draw through the center & parallel to it. draw through soe that . & .

Prop: 4

If . then the Angle is tripple to the Angle .

Prop 5

If .

Prop 6

If . that is If . then

Prop 7

If . then & .

prop 8. If . then, &c and .

prop 9 If . & or . then

Prop 10

If & . then is a side of a equall sided & angled figure. or .

prop 10

If . & a right angle & passes through the center then . And if then is perpendicular to . is the difference of the extreames and is the difference of the meanes. which given the proportionall lines may be found &c.

prop 11Pseudomesolabium

To find 2 meane proportionalls. If, . they be inscribed in the circle the : being . If twixt & two meane proportionalls are sought on the same center with the : describe & inscribe a line parallel to cutting in the point & Examine it.

prop 12

If & bisected in & bee drawne is the side of a pentagon which may be inscribed in

prop 13

If be the side of a & the side of an the arch divided in , will be the side of an to be inscribed in the circle & the arch is rightly divided by Bisecting the line .

Of Angular sections.

prop 14

If . Then or, . But the angles , are right ones and

prop 15

If the angle . or . & , are right angles then . or the triang: unequall.

prop 16.

In 2 rectang: triang: & , if the first have an acute angle submultiple to the acute angle of the 2^d triang the sides of the seacond have this proportion. Suppose the Hypoten of the first tri: be . the base . the Cathetus .

If the acute angle of the seacond triangle be to the acute angle of the first triangle in a proportion

Prop 17. If {∠} &c: & . & &c then &c & & &c. {nam} triangle & , & &c: = & sim.

Prop.18.

If &c then &c & if from ʒ to the center be drawne then &c Ergo &c

Prop 19If &c. &. are greater than the semiperiphery: & is the greatest, the least line drawn from to these points , , , . then .

prop 20 Out of the 18^th & 19^th Prop: to divide an angle into any number of points in the figure of the 18^th prop: . is the greatest of the inscribed lines : now . therfore . & . And . therfore Likewise . &

tenth &

prop 21out of the 17^th Theor.: in the figure whereof if {:} the least inscribed line . & the next line bee . then . & & & &

Prop 22 .

If . Then &c Soe that the Periph: divided into any number of points. &c. & &c.

henceProp 23. In the former scheame If . . & / therefore . & the base of the 4^th triang: & the perpendicular of the 5^t tri: & base of the 6^t triang. perpendicular of the 7^th triang base of the 8^th tri. perp: of the 9^th tri:

Prop 24:

If &c: then & & &c: & then
therefore . againe &c Therefore . & since, as Therefore . And to all the perpen dicular & transverse line . that is
( 5 ) .

Prop 24

If in the circle be inscribed the helix & touch it in the point then to the circumference.

Prop 25

If be les than halfe the circle. & . & to : then 4 times the section

Prop 26

If & perpendicular to from the angle . . then . & is the side of a heptagon

Prop 27.

If a line be cut by extreame & meane proportion the lesse segment almost is to the whole line as the diameter is to times the periphery divided by .

Prop 28

Si secetur linea per extremam & mediam proportionem erit proximè, ut tota linea plus minori segmento ad bis totam lineam, ita quæ potest quadrato sesquialterum semidiametri, ad latus quadrati circulo equalis.
linea secta sit . minus segmentum . Semidiametrum , quæ potest quadrato sesquialterum semidiametri paulo maior est quam . Radix Peripheriæ, .

Prop 28.

If . & to the side of a decagon; & parallel to then shall be almost equall to the fourth parte for is divided in extreame & meane propor in the point . & Perimeterhbkfa ; by the 27^th prop: & .

Prop 29.

If . & is divided by extreame & meane proportion in . & parallel to then is the side of a square = to the area of the circle. for by the 28^th prop: As ( .

Prop 30

If the line touch the helix in the line . & the line toucheth the beginning of it in the center & then shall bee equall to perim: . & being the Diameter: the area of the triang = to the area of the circle

Prop 31

If be a square of one revolution of an helix & the angle & through the points , , in the helix be drawne the line & through the points , in the helix be drawne . & the angle bisected by ; then shall almost touch the helix in . & it shall be soe much the nigher a touch line by how much the angles
are lesser.

Prop 32

If many Polygons be inscribed in a circle the number of theire sides increaseing in a double proportion. & theire apotomies, or the base of a tri: whose cathetus is a leg of the Polygon & hypotenusa is the Diameter (as the apotome of the Polygon is . of is &c) if the Apotome of the sides of the first Polygon be called . of the 2^d . of the 3^d . of the 4^th . of the 5^t of the Sixt . & the diameter be z And the first Polygon be . the 2^d . the 3^d . the fourth . the 5^t the sixt . the 7^th &c then
. & . & . & . & . & &c

To know how many divers ways things, whereof some of them are equall, may bee ordered. . as of . doe thus { } { } the number of changes, in order.

To know how many elections may bee made doe thus therefore there are elections in .

Propositiones Geometricae Ex Schootenij Sectionibus miscellaneis.

Sec 10

To find triangles whose sides, segments of theire bases, & perpendiculars are expressible by rationall numbers
1^st if the perpendic: is without the tri: let . . . . . . & . . . . . puting any numbers for , , & ; & may be found. then . . which reduced to the common denominator ; & that cast away. . . . . . .

In like manner if the perpendicular fall within side. . . . . . .

Also by the conjunction & disjunction of 2 triangles it may be found that . . . . . For if . . that is . . . Likewise . . . the least quantity divisible by & , being divided by them, leaves & which must multiply the bases & hypotenusas. If the perpendic: fall without the legs may be thus exprest . <14r> . . . .

July 4^th 1699. By consulting an accompt of my expenses at Cambridge in the years 1663 & 1664 I find that in the year 1664 a little before Christmas I being then Senior Sophister, I bought Schooten's Miscellanies & Cartes's Geometry (having read this Geometry & Oughtred's Clavis above half a year before) & borrowed Wallis's works & by consequence made these Annotations out of Schooten & Wallis in winter between the years 1664 & 1665. At which time I found the method of Infinite series. And in summer 1665 being forced from Cambridge by the Plague I computed the area of the Hyperbola at Boothby in Lincolnshire to two & fifty figures by the same method.                 Is. Newton

Of Refractions.

1 If the ray bee refracted at the center of the circle towards & . Then suppose . See Cartes Dioptricks

2 If there be an hyperbola the distance of whose foci are to its transverse axis as to . Then the ray is refracted to the exterior focus . See C: Dioptr

3 Having the proportion of to , or. . The Hyperbola may bee thus described.

1 Upon the centers , let the instrument bee moved in which instrument observe that & that the beame is not in the same plane with but intersects it at the angle soe that if , then . Or . Also make , i.e half the transverse diamet{er.} Then place the fiduciall side of plate in the same plaine with . & moving the instrument to & fro its edge shall cut or weare it into the shape of the desired Parabola. Or the plate may bee filed away untill the edge exactly touch it everywhere.

2 By the same proceeding Des=Cartes concave Hyperbolicall wheele may bee described by beeing turned with a chissell whose edge is a streight line inclined to the axis of the mandrill by the ∠ which angle is found by making .

3 By the same reason a wheele may be turned Hyperbolically concave the Hyperbola being convex. Or a Plate may bee turned Hyperbolically concave

Also Des=Cartes his Convex wheele may be turned or grou{illeg} {trew} a concave wheele being made use of instead of a patterne

5 In turning the concave wheele it will perhaps bee best to weare it with a stone & let the streight edged chissell serve for a patterne. And it may bee convenient to grind the stone (or iron &c ) into the fashion of a cone That may fit the hollow of the wheele . The angle of which {cone} being

9 Halving such a cone smoothly pollished within & without, by the helpe of a square set the plate perpendicular to one side the fiduciall edge being distant from the vertex the length of & if the edge of the plaine every where touch the cone, tis trew

10 The exact distance of the plate from the vertex of the cone neede not bee much regarded for that changeth onely the bigness not the shape of the figure.

[By the broken lookinglasse I find in glasse refraction, that . These are almost insensibly different from truth . Or . Or

For the Ellipsis

The former demonstrated.

Lemma. if in the Opposite Hyperbolas (one of which are to bee described) supposing . . . & terminated by the hyperbola Then is . . . . . . . And since . Or Therefore . Both points of which squared & ordered the result is . That is .

Description the 1^st demonstrated Synthetically. See that Scheame

Nameing the quantitys . . . . . . . Also , therefore That is . As in the lema

The Same demonstrated Analytically.

Nameing the quantitys, . . . . . . Supose that .

Then is . That is . Therefore the line is a Conick Section & since is greater than tis an Hyperbola, which that it may bee the same with that in the lemma, Their correspondent termes are to bee compared together & soe I find that . & by the 1^st equation . Or . that is . by the 2^nd . And by substituting into the place of And ordering it tis . Or . Therefore if I take . & . then shall bee the Hyp^erbola desired Q:E:D.

The 2^d 3^d 4^th & 5^th Propositions are manifest from this

Instead of the 6^th & 7^th Descriptions which are false use those

6 Draw 2 concentrick circles ( & ) with the Radÿ & . Then from the comon center draw 2 lines & at the given angle of then draw a line from by the end of the Rad & to the intersection of that line with the circle draw & so the angle of is found.

Or which is the same make . & then if that cone is sought the angle being given, make . Then is . & soe the is knowne & also , & . But if the of the section is sought the cone being given than make . And it will bee . & soe is given also . &

In general observe that in any cone cut any ways . & .

7. DesCartes his wheele thus described cut by any plaine produceth one of the Conick=Sections.

Description the 6^th Demonstrated. Synthetically.

Call, . . . . . . . . . . Therefore . by ordering the result of . which is like that in the lemma.

The 7^th Proposition may be easy ly demonstrated after the same manner

If the two equall cones intersect the one the other soe that their intersection shall bee one of the Conick sections as they had each beene intersected by the plane .

To describe the Parabola (& other figures after the same manner) pretty exactly.

Take a squire , soe that (for then the circle described by will bee as crooked as the Parabola at the vertex ). Divide the other leg of the Squire into any number of points, Then get a plate of Brasse &c: streight & eaven. And taking one point for the vertex of it & another point for the Squire to moven soe that , & weareing away the edge of the plate untill (the Squire being erected) . the squire touching the plate at . thus shall the edge become Parabolicall. the describe a circle it may bee knowne when . Instead of the leg a circle may be used Demonstracon. . . . . & . Q.E.D.

Another description of the Parabola with the compasses. Make . Make & . Make , & then shall be a point in the Parabola.

Another. Make . & the point shall bee in the parabola. This like the first by calculation may bee made use of in other lines.

The manner whereby any kind of little lines may be described very accurately. And that the same Instrument serve for all lines (though never so small) differing in quantity but not in quality.

Make the plate of the figure required (by some of the former meanes) the larger the better. Then hold the streight steele staffe against the center & {roule}{route} it to & fro it shall grind into the same figure but soe much lesse as is lesse than .

Soe if the glass bee fastened upon the mandrill , it may be ground acording to the sollid figure by the helpe of a stick of steele (as a cone) whose cuspis is in the hole upon which it is moved as on a center. when the cone leanes uppon the vertices of & it must be perpendicular to the mandrill . Perhaps it may be convenient to cause the cone to turne about its axis. Or it may bee better instead of the nutt at with a hole in it to make a sharpe pointed nutt, & instead of the cone to make use of a broad plate to cover , & & move every way upon them

Another way to describe lines on plates

Suppose the plate bee , whose edge is to be made into the fashion of a given crooked line supposes is its vertex & that a circle described with the Radius would bee as crooked as the given line at its vertex. Againe suppose two streight rulers & to bee very trew & steddyly fastened together which must a very little incline the one to the other, soe as that being produced they would meete at . Then are the lines , & given.

Suppose then the point in the crooked line is to bee found then is given by supposition, & consequently (supposing to bee a tangent) . . . . . . . & (if ) then is . being thus found, supposing that , then I take . that is . haveing thus found the point lay the plate twixt the two rulers so that the point of it, fall upon the point then should the line touch the plate in . But note that .

In both telescopes & microscopes tis most convenient to have a convex glasse next the eye for by that meanes the angle of vision will bee much greater than it will bee with a concave one (though both doe magnifie alike). If the convex glasse be Hyperbolicall (&c) make it soe bigg that the penecilli may crosse in the pupill; that is, the exterior focus will be as far distant from the vertex as the eye is. let the glass bee as thinn as may bee that the eye bee not too far from the vertex that it should bee about as thick as the distance of the interior focus from the vertex.

And by this meanes also, (the focus of the objectglasse being within the telescope twixt the glasses) there may bee placed at that focus the edge of <30v> a steele ruler accurately divided into equall parts (to measure the diameters or distance of starrs &c) which should bee soe made that by a pinne or handle it may be placed in any posture & in any parte of the focus, without otherwise altering the Telescope in observations.

Note that were not the glasses faulty they would not onely magnify objects but render vision more distinct; each of the penicilli passing through (perhaps but) the 10^th, 20^th or 100^th parte of the pupill must bee more exactly refracted to one point of the Tunica Retina than in ordinary visio in which each of the penicilli spreads over all the pupill.

Note also that that the glasse may be ground Hyperbolicall by the , if it turne on the mandrill whilst turnes on the axis being inclined to it as was shewed before. If the edge bee not durable enough, inough instead thereof use a long small cilinder: which I conceive to bee the best way, of all. For a Cilinder of all sollids is most easily made exact (being turned, as in the figure, by a gage untill its thicknesse bee every where equall). 2 the Cilinder may bee made to slip up & downe & turne round whereby it will not onely grinde the glase crosse {wife}{wise} to take of all hubbes, but also the glasse & cilinder will grinde the one the other truer & truer. All the difficulty is in placing the axis perpendicular to the Mandrill & vertex to vertex, which yet may bee done exactly severall ways. & untill then the glasse & Cilinder will not fit. & should the axis not intersect the glasse would bee still Hyperbolicall except a point at the vertex of it. The same instrument may also serve for severall glasses onely making longer or shorter. Let the Cilinder han{g} over the glasse.

To Grinde sphæricall optick Glasses

If the glasse is to bee ground sphærically hollow: naile a steele plate to the beame , on the upper side: In which make a center hole for the steele point of the shaft : to which shaft fasten a plugg of stone or leade or leather &c: (with which you intend to grinde the glasse ): which shaft & plugg being swung to & fro upon the center will grind the glasse be sphærically hollow.

The manner whereby glasses may bee ground sphærically convex may appeare by the annexed figure (being the former way inverted). Also the plugg , in the figure, is ground sphærically

But if this way bee not exact enough yet hereby may bee {grownd}{ground} plates of mettall well nigh sphæricall, And by those plates may bee ground glasses after the usual manner; If a circular hoope of steele bee about the edge of the glasse to keepe it from grinding away at the edges faster than in the middle.

But the best way of all will bee to turne the glass circularly upon a mandrill whilest the plate is steadily rubbed upon it or else <31v> to turne the plate upon a mandrill whilest the glasse is rubbed upon it or let sometimes the one, sometimes the other bee turned.: & by this meane{s}{} they will either of them weare the other to a truely sphericall forme. but however let there bee a hoope or of some mettall which weares more difficulty then glasse to defend the glasse from wearing more at its edges then in the middle. Perhaps it may doe well first to weare the plate sphæricall by the hoope alone without the glasse.

The same meanes may bee used for grinding plaine glasses.

Let not an object glasse bee ground sphærically convex on both sides, but sphaerically convex on one side & plane or but a little convex ~ on the other, & turne the convexest side towards the object.

If the Glasses of a Telescope bee not truely ground Theire errors may bee thus found.

Because an error is much more easily discernable in the object glasse than in the eye glasse let us first suppose the eye glasse to bee ground true towards its center, (tis exact enough if it be sphericall, & not Hyperbolicall), & so wee may find & rectifie the errors of the object glasse.

First make a thin plate of brasse & in the center of it a Small hole (whose diameter perhaps may bee about the 50^th or 100^th parte of an inch. With which plate cover the eye glass the center of it respecting the center of the glasse.

Secondly make two other plates the one with two holes as neare to its edge as may bee their{e}{} distance being about the 5^th parte of an inch or lesse, & the other with one hole close to the midst of its edge. Let the diameters of these 3 holes bee about a 20^th parte of an inch or lesse. And theire edges must bee true that they may slide one upon another, & that not let the suns rays passe through, to which purpose make them oblique. with these two plates cover the object glasse (first stopping the hole of the holes of the other plate respecting the center of the glasse & looke at a stare (or the edge of the sunne &c) & if the object appeare double (like two starrs &c) make the Tube longer or shorter until it appeare single. Then open the hole of , & the plate being fixed, slide the plate up & downe still looking at the starre, When then appeares <33r> but one starre that part of the glasse under the hole of is truely ground in respect of the 2 parts of the glasse under the two holes of . But {no} when the starre appeares double. And the position of the starre caused by the hole of in respect of the starre caused by the holes of , shews which way the glasse under the hole of is erroneously inclined; the distance of the two starres giving the quantity of that error.

Thus the errors of the object glasse bein{g} found in every place of it they may bee all rectified, & found againe, & againe rectified, untill they almost or altogether vanish.

Then may the eye=glasse bee rectified much after the same manner, in every parte of it, & if it bee necessary the object glasse may bee againe rectified & againe the eye=glasse untill the Telesope bee as perfect as the workeman can make. Whome perhaps experience may teach by this & the former rules to make telescopes as perfect as men can hope to make them.

These glasses may also bee rectified whilst on the Mandrill by observing the images made by reflection from the vertex & all other points of the glasse with proportion they have one to another & how much they are longer than broader in one place then another. &c.

of the method of infinite series

I Newton

Theoremata varia. Circa angulorum æqualitates

si ang & bisecentur a rectis et et ducatur quævis . Erit
1. Euclid 6 3 2. . Scho{o}te de {concis} {æqu{is}} 3. posito .

Si in angulo quovis inseribantur æquales , , , , , , &c anguli erit angulus duplus tripl, quadr quint, sext, sept. oct &c. Horu vero angulorum posito radio sinus erint , &c cosinus , , &c. Ergo si , & erit . . . &c

In Generall.

. . &, .

VII. 14

{illeg} Geomet. {illeg}. Frn Vieta Schooten {illeg} July 4^th 1699 {illeg}

To find the sume of the squares cu{bes} &c. of the rootes of an equation

If , , , , , &c be the rootes of the equation . then is &c &c { } &c: .

. . . . . . . . . .

Ad constructionem Canonis angularis.

. . et . . . . . .

If radius. Then

Suppose . . Then

As on the other leafe excepting some signes have changed.

If . .Then

Of Angular sections

Suppose . . & . & that the arches , , are equall. By the following equations an angle may bee divided into any number of partes.

This scheame is the former inversed.

Suppose the perifery to bee & the whole perifery to bee . The line subtends these arches. . . . . . . . . . . . . . &c: All which are bisected, trisected, quadrisected, quintusected &c after same manner. As for example

The rootes of the equation . are 3. The first whereof subtends the arches . . . . . . &c. The second subtends the arches . . . . . &c. The 3^d . . . . &c.

Soe the rootes of the equation ~ , doe the first subtend the arches . . &c: the 2^d . . . the 3^d . . &c. the 4^th . . &c the 5^t . . . &c.

Hence may appeare the reason of the number of rootes in these equations & that the points of the circumference to which they are extended æquidistant. & by the lower scheme may bee known which rootes are affirmative & which negative.

The numerall cöefficients of the afforesaid equations may bee deduced from this progression (if .) &c. As if . the progression . And the coefficients .

1663 _/4 January.

All the parallell lines which can be understoode to bee drawne uppon any superficies are equivalent to it, as all the lines drawne from to may be used instead of the superficies

If all the parallell lines drawne uppon any superficies be multiplied by another line they produce a Sollid like that which results from the superficies drawne into the lame line as if either all the lines in the superficies or if the superficies be drawne into the line they both produce the same sollid whence All the parallell superficies which can bee understoode to bee in any sollid are equivalent to that Sollid. And If all the lines in any triangle, which are parallell to one of the sides, be squared there results a Pyramid. if those in a square, there results a cube. If those in a crookelined figure there results a sollid with 4 sides terminated & bended according to the fashion of the crookelined figure{.}

If each line in one superficies bee drawne into each correspondent line in another superficies as in , & if . . . &c. they produce sollid whose opposite sides are fashioned by one of the superfic as Sollid . where all the lines drawne from to are equall to all the correspondent lines drawne from to . & those drawne from to are equall to the correspondent lines drawne from to .

Theorema. 1

If in the Circle there be inscribed any Poligon with an odd number of sides, & from any point in the circumference there bee drawne lines , , , , to every corner of the Polygon: the summ of every other line is equall to the summ of the rest, . & soe are their cubes . unless the figure be a Trigon

Theor 2

If from the points of the Polygon then bee drawne perpendicular , , , , to any Diameter point: the summe of the Perpendiculars on one side the Diameter is {equall}{equal} to their summe on the other . & soe is the summe of their cubes (unlesse when the figure is a Trigon), . & of theire square cubes (except when the figure is a Trigon or Pentagon. &c.

Theor 3

If the 2 circles (fig 1 & fig 2) be equall with like Poligo{illeg}{ns} inscribed, & in fig 1 be assumed double to in fig 2. then are all the other corresponding lines in fig 1 double to those in fig 2 viz , , , .

To square the Parabola

In the parabola suppose the Parameter . . . & . Now suppose the lines called doe increase in arithmeticall proportion all the 's taken together make the superficies which is halfe a square let every line drawne from to be square & they produce a Pyramid equall to every . which if divided by there remaines equall to every equall to every or all the lines drawne from to equall to the superficies equall to a 3^d parte of the superficies & the superficie .

Otherwise. suppose . . . & the lines increasing in arithmeticall proportion every is equall to 4 times the superficies but if every be squared they produce a pyramid equall to . wherefore every equall to every equall to the superficies drawne into & to as before.

&

To Square the Hyperbola

In the Hyperbola . suppose . . {} . . & . . In which equation Every taken together is equall to the triangle equall to & every taken together is a pyramid . Every taken together is equall to the superficies If then . every is equall to the solid . If the angle is a right one & if that is if the triangle . every will be equall to the sollid Joyne these two sollids together as in .

Againe Suppose every taken together to be equall to the superficies , the line squared is every composeth a Sollid like an eighth parte whereof (which is equall to every ) being like ; will be equall to & . whence the convexe superficies of the figure will fitly joyne with the concave superficies of the figure . If every is equall to the superficies , every shall be equall to the triangle . every every & therefor the Sollid . Joyne the Sollid to & there resulteth from which againe substract & there remaines the sollid & which substract from the sollid & there remaines which being divided by . there remaines to the superficies

The squareing of severall crooked lines of the Seacond kind

In any two crooked lines I call the Parameter or right side of the greater. . but of the lesse Transverse side . the right axis as or . Transverse axis as , or .

Suppose in the Parab: : . & in : . . . . . Or . if . . make . . . . & therefore the square of the crooked line (when the line is supposed too close with the line ) whose nature is exprest by the foregoing equation.

. . . . . . or,

3 . . . . .

. . . . Or .

. .

4 In the Parabola . . . . { } . . Since all all . . therefore .

. . . . Then shall : be the axis of gravity in &

In the 1^st figure.

. . . . or . Suppose the swiftnesse of to the swiftnesse of . . .

Fig 2^d. 3^d.

. its swiftnesse its swiftness . . . .

Fig 4

motion of the point from motion of the point from increasing of increasing of motion of motion of . &c as before.

These are to find such figures , , as doe equiponderate in respect of the axis .

Reasonings concerning chance.

If

1 If is the number of chances by one of which I may gaine , & those by one of which I may gaine , & those by one of which I may gaine ; soe that those chances are all equall & one of them must necessarily happen: My hopes or chance is worth . The same is true if , , signify any proportion of chances for , , .

2. If I bargaine for more than one chance (viz: that after I have taken the gaines by my first chance, from the stake ; I will venter another chance at the remaining stake &c) my second lott is worth . My third lot is worth . My Fourth lot is worth . My Fift lot is worth . My sixt lot is worth . &c

As if 6 men cast a die soe that he gaines who throws a cise first: since there is but one chance to gaine & 5 to gaine nothing at each cast, I make . & . Therefore by the <90v> The first mans lot is worth The seconds is worth . The thirds is worth . The fourths is The fifts lot is worth . The Sixts lot is . &c. Soe that their lots are as .

Soe that if I cast a die two or more times tis to that I cast a cise at the first cast & to that I throw it at two casts, & to that I cast it at thrice, & to that I cast it once in 4 trialls, & to that I cast it once in 5 times. &c

3. If I bargaine to cast severall sorts of lots successively at the same stake the valor of each lot is thus found viz: The first prop: gives the valor of the first lot; which valor being destructed from the stake, the remainder is the stake of the 2^d lot which therefore may bee also found by the first prop: &c.

As if I gaine by throwing at the first cast, or at the 2^d or at the 3^d &c with two dice. Since at the first cast there is but one chance for (viz ) & for nothin{g} Therefore its valor is (by Prop 1). & the stake for the 2^d cast is . Now since there are two chances for it (viz: ⚅⚄ & ⚄⚅{)} & for at the 2^d cast therefore its valor is . as the stake for the 3^d lot is for which there are 3 chances (viz ⚄⚄, ⚅⚃, ⚃⚅) & for nothing Therefore its valor is .

4 If I bargaine with one or two more to cast lots in order untill one of us by an assigned lott shall win the stake : Since the chances may succede infinitly I onely consider the first revolution of them The valor of each mans whole expectation being in such proportion one to another as the valors of their lots in one revolution. & the valors of each mans first lot being to the valor of his whole expectation as the summe of the valors of their first lots to the stake .

As if I contend with another that who first throws with 2 dice shall have , I haveing the dice. My first lot is worth (by prop 1), The 2^d his first lot is worth . And . for the two first lots make one revolution because I have the same lot If I throw a 2^d time that I had at the first. Therefore is my interest in the stake.

If our bargaine bee soe that there is some lott at the beginning of our play which returnes not in the after revolutions, detract the valor of those irregular lotts from the stake & the rest shall bee the stake of the lots which follow & revolve successively. As if I contend with another that who first casts must have , onely I have {the} first cast for . My first lot is worth . & the stake for our after throws is . his firts lot being . & my next lot . soe that his share in the stake is to mine as . Soe that my share in it is . To which adding the valor of my first lot viz: , the summe is , my interest in the stake at the begining.

5 If the Proportion of the chances for any stake bee irrationall the interest in the stake may bee found after the same manner. As if the Radÿ , , divide the horizontall circle into two points <91v> & in such proportion as to . And if a ball falling perpendicularly upon the center doth tumble into the portion I winn : but if into the other portion, I win . my hopes is worth .

Soe if a die bee not a Regular body but a Parallelipipedon or otherwise unequall sided, it may bee found how much one cast is more easily gotten then another.

Soe that the facility of the chances the stake belonging to each chance being knowne the worth of the lott may bee ever found by the precedent precepts. And if they bee not both immediatly known they must bee sought before the valor of the lott can bee found.

As if I want two games at Irish & my adversary three to win , & I would know my interest in the stake my first lot can gaine me nothing but the advantage of another lot, & therefore to know its vallue I must first find the value of that other lot &c. First therefore if wee each wanted one lot to win our interest in it would bee equall viz my lot worth . Secondly If I want one game & my adversary two, & I gaine the next game then I gaine but if I loose it I onely gaine an equall lot for at the next game which is worth , Therefore my interes{t} in the stake is . Thirdly If I want one game & my adversary three & I gaine the next game I get ; but if I loose it, then I want one game & my adversary but two, that is I get : Therefore (there being one chance for & one for ) my interest in the stake is . Fourthly If I want 2 games & my adversary 3; & I win I get . but if I loose I get for our chances <92r> will then bee equall; Therefore my interest in the stake is . Soe if I want games & my adversary my interest in is . If I want two and hee , it is . If I want and hee it is . If I and hee , it is: . If I and hee it is . If I and hee it is . If I and hee , it is: . (The like may bee done if 3 or more play together. (as if one wants one game, another 3 a third 4: Their lots are as . &c.) As also if their lots bee of divers sorts.)

By this meanes also some of the precedent questions may bee resolved. as if I have two throws for a cise to win , with one die; If I have missed my first lot already I have at my second cast five chances for nothing. & one for . therefore that cast is worth . Soe that in my first cast I had five chances for & one for , which therefore (with my 2^d cast) is worth . That is tis to that I cast a cise once in two throws. as before

By this meanes also my lot may bee known if I am to draw cards of severall sorts out of cards of each sort.

Or if out of two white & black stones I am blindfold to chose a white & a black one.

Equation

An equation given; if both , , have divers dimensions, try if the roote of one of may be extracted: & If a quantity wherein is not is divided by in the line equall to . that crooked cannot be squared.

The line is a Parab. . . . . . . . . . . or supposeing y. & . which shews the nature of the crooked line . now if . then . for supposeing moves uniformely from , moves from with motion decreaseing in the proportion that the line doth shorten. Suppos & . . then . suppose . then . Or . Or . Or suppose . then . Or Or . Or, if . . &
. Or . . . . . . . . . <93v> <94r> .

Let . . .

. . . . . .

To square those lines in which is onely

If is in but one terme onely of the Equation (as . or, ) resolve the Eq: into the proport (as . or, .) If the line hath Assymptotes . .

. divided by it produceth.

By the Squares of the simplest lines to square lines more compound. 1^st those whein .

find the valor of . If the number of the termes in the denominator thereof be neither &c. the line cannot be squared If it have but one terme tis squared by finding the square of each particular terme in the valor of & then adding all those squares together. Example 1^st. . & . Then makeing equall to each particular terme. . or whose square is . & . whose square is Add these 2 squares together & they (viz: ) are the square of the line . Againe . Or . then disjoynting the valor of . . . Or , whose square is . , whose square . , whose square . which 3 squares (viz ) taken together are the square sought for. And these lines may bee ever squared unless in the valor of there bee found , , , &c. for the Squareing of that line depends on the squareing of the Hyperbola. As in the line .

Secondly. If it have 3 termes See if it may be reduced to dimensions by adding or subtracting a knowne quantity to or from . Example. . which (makeing ) is thus reduced
. Or

. . .
. & . .

. . . . . An equation expressing the nature of the line .

. .

. . . .

.
.

. . . . . .

. . .

. . . . . . . . . { } . . . . &c: . . . . which skewes the nature of another crooked line that may be squared.

. . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . .

This table shews the distance of any two notes As the distance of & is , or a third, or halfe not{es.} Of & tis a fourth, or halfe notes. of & tis halfe notes, or greater than a fifth ♭, by halfe notes &c.

. . . . . . . . .

.
.





.

By the helpe of concordant notes all the notes in the Gam ut may bee thus tuned viz:

First tune the eighths, &c.

Seacondly tune fifts to them both above them & .

Thirdly tune thirds to them both above them & below them .

Fourthly from each rise a fift for & fall a fift for .

Fiftly from rise a fift for & fall a fift for .

Sixtly from rise a fift for & from fall a fift for .

Seaventhly from each rise a fift for . The rest as are supplyed by eighths viz to &c.

November 20. 1665.

. . . . .

. . . . . . . . . . . perhaps is better than .

By this table may bee knowne the of any two notes whither a {trew} second of the lesse, second, third the lesse, a third fourth &c: As to know the distance twixt & I follow the pricked stroke from to or from to where I find it crossed by a black crooked line & against it, written, therefore I conclude & distant a true fourth.

And Thus to find the distance of & I follow the prick line from the top to the right hand side thence to the bottom thence towards the left hand side untill I come {over} . Or (which is the same) I follow the prick{t} line from the top to the left hand side thence to the bottom , thence toward the right hand side untill I come just over , where I find the pricked line to be crossed by a stroke & against it to bee written on the upper line , on the lower therefore tis exactly. But if it

. . . .

. . . . . . . . . .

The 3 meanes are best there being an imperfect fift in the outward extreame & a tritonus in the inmost.

In the Hyperbola . suppose . . a secant. . .
. . . . . which equation continues the nature of the crooked line . Now supposeing the line always moves over the same superficies in the same time, it will increase in motion from in the same proportion that it decreaseth in lenght & the line will move uniformily from (), soe that the space . suppose . . . & .

Modi

In the order of the musicall tones the 2 halfe notes may not be together 1^st because every note would then bee distant 3 tones from some other which is most ungratefull Secondly whole notes ought to bee interposed to moderate their harshnesse. Thirdly since there must bee a Fift to the ground: these notes must bee either next the ground or its Fift which would make them harsh & that {} could not gradually passe to or from them.

Neither ought they to be distant but one tone for the second reason {afforesd} & because they will bee more consonant by the absense of more 3 tones &c if they be distant 2 tones yet perhaps they may not bee wholly uselesse. See the last modes.

A catalogue of the 12 Musicall modes in theire order of gratefulnesse.

suppose the line last found to be . . . . . . . . . to find at what point : . . . . . . . . . which shews the nature of the line . & or . suppose . . . .

. Suppose . . . . . . . . . . . . . . . .

tone max; medî: minimus. , , tone {maj mi.}
.

. hath 8 Fifts

Suppose againe the last line whose nature is comprised in this equation . . . . . . . to find where
. . . . . . . .

. . . .

1^st. . . . . . . . . . per sup. et Modus harum vocum respectu fundamenti

2^d. . . . . . . . And if . . Then & the voyces in respect of their {ground} are best

If in the 1^st case . then If in the 2^d case then

Likewise supposing the line . . . . . . . &c whence supposeing to be a line increasing in arithmeticall proportion from the quantity of the line untill it be as long as . the superfices resulting out of &c . . . . . &c

1 Of the Key or Ground sound. Secondly, Of its Eighths. Thirdly, of their divisions into Fifts & Fourths Sixts & Thirds, illustrated by the division of a corde. Fourthly, The order of the concords in respect of gratefulnes deduced thence & from other considerations. Fifthly the degrees deduced thence & of the proportion of the concords & degrees i.e. the logarithmes of their strings. 6 Of the various ordering of the degrees & distance of the halfe notes , the keys fift being onely stable 7 Of the moodes ariseing thence & their dignity; explained by one line, &c. Eighthly, How the tones major & minor are best ordered in every Moode. Ninthly of passing from one moode to another explained by 3 lines 10 How the notes major and minor to be
ordered for that purpose.

in the Hyperb: . . .

. . . . . . : . . .