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SP x

Cas. 1. Distinguatur enim tempus in particulas aequales et quam minimas, et agat vis W non continuò sed singulis temporis particulis semel. Sit autem T velocitas Lunae [in] Pante impulsum vis W ibi factum et t incrementum [velo]citatis ex impulsu et L latus rectum Orbis Lunaris ante [impulsum]. Et quoniam area quam Luna radio ad Terram [ducto singulis temporis particulis aequalibus describit, sit ante impulsum ad eandem aream post impulsum ut T ad T'+t, et latus rectum (per Prop. xiv. Lib. 1. Princip.) sit in duplicata ratione areæ, erit (per Lem. Lib. II. Princip.) T + 2t

2t L seu L+ L latus rectum post impulsum. Est autem (ut in T

T

SP x L Lemmate superiore)

longitudo PF qua Luna distabat

2SP + 2PK-L ab umbilico superiore ante impulsum; et propterea cum situs lineae PF, si modo excentricitas SF infinitè parva sit, ex impulsu illo nil mutetur, ideoque PK maneat eadem quae prius et solum L mutetur, si producatur PF ad $ ut sit $ umbilicus superior post impulsum;

T + 2t

L

T erit Po aequalis

De hac longitudine subdu

2t 2SP+2PK-L-L T

SP x L catur longitudo ipsius PF superius inventa, nempe 28P+2PK - L' et interea in utraque pro 2SP + 2PK scribatur 2L & manebit

4t

4t differentia aequalis SP seu , SP.

SP. Unde longitudo per

T - 2t pendiculi pg quod in diametrum QR ab umbilico o demittitur, erit 4t

PE. Jam vero in Lemmate superiore, velocitas quam vis V T impulsu unico generare potest, est ad velocitatem Lunae ut lineola pG quam Luna vi impulsus illius dato tempore describere posset ad lineolam Pp quam Luna velocitate sua data T eodem tempore describat, id est ut Ff ad PF. Ideoque si velocitas prior nominetur S erit

28 x PF Ff aequalis

ob angulum FPf anguli GPp duplum, et per

T pendiculum fh quod ab umbilico f in ellipseos axem QR demittitur

28 aequale EF. Proinde cum angulus ASF sit ad angulum FSf ut

T og ad fh, et angulus FSf ad angulum PSp ut VSE ad P x OS,

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i This should be pg ad Pp.

erit angulus PSF ad angulum PSp, hoc est motus Apogaei a vi W

4t

28 genitus ad motum medium Lunae ut PE ad EF et V SE ad

T

T Px OS conjunctim, id est (ob aequales EF ad SE et proportionales t & S, W& V) ut 2W x PE ad P x OS. Q.E.D.

COROL. Obtinet etiam Propositio quam proxime ubi [quam minima sit] excentricitas etiamsi non sit infinitè parva.

SECTION 1.

MATHEMATICS.

I. EARLY PAPERS BY NEWTON. (Holograph.)

1. Extracts by Newton

From Hooke's Micrographia,
From the History of the Royal Society,

From the Philosophical Transactions.

Notes of some Mines in Derbyshire and Cardiganshire. 2. Scraps and Extracts made by Newton, including two little notes on tangents and musical semi-tones.

3. A tract in English written in 1666, entitled "To resolve problems by Motion.” Also short tracts entitled De Solutione Problematum

per

Motuin.
De Gravitate Conicarum.

Problems of Curves.
4. Calculation of the Area of the Hyperbola.
5. On the Laws of Motion.

On the Laws of Reflection.

On Motion in a Cycloid. 6. Problems in Geometrical Optics.

II. ELEMENTARY MATHEMATICS. (Holograph.) 1. Observations on the Algebra of Kinckhuysen.

2. The first Ten Propositions of the 2nd book of Euclid, succinctly enunciated and demonstrated.

3. Theorem on the Area of a Triangle.

4. Trigonometria succinctè proposita et nova methodo demonstrata a St Joanne Hareo Arm.

A few MS. leaves, containing Compendium Trigonometriae. It includes Spherical Trigonometry. Intended for learners. 6. Table of sines to every

half degree.

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III. FLUXIONS. 1. Transcript of a Tract on Fluxions said to have been written by Newton in November, 1666.

2. Tract relating to the History of Fluxions, transcribed from one which was probably written by Jones.

3. Part of Newton's method of Fluxions and Infinite Series, with a fragment of the same treatise. (Holograph.)

4. Part of a Tract on Fluxions.

5. Some Propositions in Fluxions. ["I think this fragment very proper to be published.” Horsley, Oct. 22, 1777.]

6. Analysis per quantitates fluentes et eorum momenta.
7. Method of Fluxions and Infinite Series.
8. On the solution of Fluxional Equations.
9. Fluxions applied to Curves.

10. Propositions in the Method of Fluxions (dotted letters employed).

11. Propositions in Fluxions (dotted letters employed).

12. An early paper on deducing the subnormal in a curve from a given rational relation between 3 and y, and the converse operation.

13. Fragments on Fluxions.

14. Method of Curves and Infinite Series, and application to the Geometry of Curves. Complete all but the 1st leaf.

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IV.

ENUMERATION OF LINES

(Holograph.)

HIRD

1. An early copy. 2. A later copy.

3. On the curves of the third order, produced by the projections of the Parabola Neiliana.

4. Fragments concerning lines of the third order, and some mistakes of Descartes ["not worth publishing.” S. Horsley, Oct. 23, 1777].

V. ON THE QUADRATURE OF CURVES. (Holograph.)

1. A copy which appears to be pretty complete.
2. A fragment on the same subject.
3. Scattered papers on the same subject, in great confusion
4. Another fragment on the same.

5. Note on Quadrature of Curves, intended as a Supplement to Section 10 of Book I. of the Principia.

6. Fragment on the Quadrature of Curves whose equations consist of but three terms.

VI.

PAPERS RELATING TO GEOMETRY. (Holograph.)

1. De Problematum resolutione Synthetica. 2. Geometria. Liber 1.

Liber 1. A fragment. 3. Geometry, a fragment on Porisms. 4. Analysis Geometrica.

Regula Datorum. 5. Newton's Regula Fratrum, 6. Fragment relating to Curves. 7. Geometria Curvilinea and Fluxions. 8. Scraps containing Propositions in Geometry; viz. : (a) To describe a Conic Section through five given points;

and (6) To describe a Conic Section passing through two points

and touching three given straight lines. 9. Tract on the construction of Equations, unfinished. 10. On the Properties of Curves. 11. Part of a Treatise on Geometry (in Latin). 12. De Compositione Locorum Solidorum. 13. Solutio Problematis Veterum de Loco Solido.

14. Fragments relating to the writings of the Ancients in general, but especially to the Porisms of Euclid, and the Loci of

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