[1] Philosoph. Nat. princ. math. L. iii. introduct.

[2] Nov. Org. Scient. L. i. Aphorism. 9.

[3] Nov. Org. L. i. Aph. 19.

[4] Ibid. Aph. 25.

[5] Aph. 30. Errores radicales & in prima digestione mentis ab excellentia functionum & remediorum sequentium non curantur.

[6] Aph. 38.

[7] Ibid.

[8] Aph. 39.

[9] Aph. 41.

[10] Aph. 10, 24.

[11] Aph. 45.

[12] De Cartes Princ. Phil. Part. 3. §. 52.

[13] Fermat, in Oper. pag. 156, &c.

[14] Nov. Org. Aph. 46.

[15] Aph. 50.

[16] Ibid.

[17] Aph 53.

[18] Aph. 54.

[19] Aph. 56.

[20] Aph. 55.

[21] Locke, On human understanding, B. iii.

[22] Nov. Org. Aph. 59.

[23] In the conclusion.

[24] Nov. Org. L. i. Aph. 59.

[25] Ibid. Aph. 60.

[26] Ibid. Aph. 62.

[27] Aph. 63.

[28] Aph. 64.

[29] Aph. 65.

[30] See above, § 4, 5.

[31] Nov. Org. L. i. Aph. 69.

[32] Ibid.

[33] Ibid. Aph. 109.

[34] Book III. Chap. iv.

[35] Book I. Chap. 2. § 14.

[36] Ibid. § 85, &c.

[37] See Book II. Ch. 3. § 3, 4. of this treatise.

[38] See Book II. Ch. 3. of this treatise.

[39] See Chap. 4.

[40] At the end of his Optics. in Qu. 21.

[41] See the same treatise, in Advertisement 2.

[42] Nov. Org. Lib. i. Ax. 105.

[43] Princip. philos. pag. 13, 14.

[44] Princ. Philos. L. II. prop. 24. corol. 7. See also B. II. Ch. 5. § 3. of this treatise.

[45] How this degree of elasticity is to be found by experiment, will be shewn below in § 74.

[46] In oper. posthum de Motu corpor. ex percussion. prop. 9.

[47] In the above-cited place.

[48] In the place above-cited.

[49] These experiments are described in § 73.

[50] Book II. Chap. 5.

[51] Chap. 1. § 25, 26, 27, compared with § 15, &c.

[52] Book II. Chap. 5. § 3.

[53] See Euclid’s Elements, Book XII. prop. 13.

[54] Archimed. de Æquipond. prop. 11.

[55] Ibid. prop. 12.

[56] Lucas Valerius De centr. gravit. solid. L. I. prop. 2.

[57] Idem L. II. prop. 2.

[58] § 25.

[59] § 27.

[60] Pag. 65, 68.

[61] § 23.

[62] § 20

[63] § 17.

[64] § 27.

[65] Hugen. Horolog. oscillat. pag. 141, 142.

[66] See Hugen. Horolog. Oscillat. p. 142.

[67] Princip. Philos. pag. 22.

[68] Chap. 1. § 29.

[69] Princip. Philos. pag. 25.

[70] § 71.

[71] See Method. Increment. prop. 25.

[72] Lib. XI. Def.

[73] Chap. 2. § 17.

[74] See above Ch. 2. § 17.

[75] From B II. Ch. 3.

[76] Prin. Philos. pag. 7, &c.

[77] See Newton, princip. philos. pag. 9. lin. 30.

[78] Princip. Philos. pag. 10.

[79] Renat. Des Cart. Princ. Philos. Part. II. § 25.

[80] Ibid. § 30.

[81] § 85, &c.

[82] Princip. Philos. Lib. I. prop. 9.

[83] § 92.

[84] Ch. II. § 22.

[85] Viz. L. I. prop. 30, 29, & 26.

[86] Ch. II. § 21, 22.

[87] viz. His doctrine of prime and ultimate ratios.

[88] § 57

[89] § 3.

[90] Ch. 2. § 22.

[91] § 12.

[92] Ch. 1. sect. 21, 22.

[93] Elem. Book I. p. 37.

[94] § 12.

[95] Ch 1 § 24.

[96] Ch 2 select. 17.

[97] Newt. Princ. L. II. prop. 2; 5, 6, 7; 11, 12.

[98] Prop. 3; 8, 9; 13, 14.

[99] Prop. 4.

[100] PrÆlect. Geometr. pag. 123.

[101] Newton. Princ. Lib. II. prop. 10.

[102] Newton. Princ. Lib II. prop 10. in schol.

[103] Torricelli de motu gravium.

[104] Ch. 2 § 85, &c.

[105] Newt. Princ L. II. sect 6.

[106] L. II. sect. 4.

[107] See B. II. Ch 6. § 7. of this treatise.

[108] Lib. I. sect. 10.

[109] De la Pesanteur, pag. 169, and the following.

[110] Newton. Princ. L. II. prop 4. schol.

[111] See his Tract on the admirable rarifaction of the air.

[112] Book II. Ch. 6.

[113] Princ. philos. Lib. II. prop. 23.

[114] Book I. Ch. 2. § 30.

[115] Princ. philos. Lib. II. prop. 23, in schol.

[116] Princ. philos. Lib. II. prop. 33. coroll.

[117] Lib. II. Ch. 5.

[118] Ibid. Prop. 35. coroll. 2.

[119] Ibid. coroll. 3.

[120] Vid. ibid. coroll. 6.

[121] In § 2.

[122] Princ. philos. Lib. II. Prop. 35.

[123] Ibid.

[124] Id.

[125] h. 1. § 29.

[126] Princ. philos. Lib. II. Prop. 38, compared with coroll. 1 of prop. 35.

[127] L. II. Lem. 7. schol. pag. 341.

[128] Lib. II. Prop. 34.

[129] Lib. II. Lem. 7. p. 341.

[130] Schol. to Lem. 7.

[131] Prop. 34. schol.

[132] Ibid.

[133] Ibid.

[134] Book II. Ch. I. § 6.

[135] Vid. Newt. princ. in schol. to Lem. 7, of Lib. II. pag. 341.

[136] Sect. 17. of this chapter.

[137] See Princ. philos. Lib. II. prop. 34.

[138] Vid. Princ. philos. Lib. II. Lem. 5. p. 314.

[139] Lemm. 6.

[140] Ibid. 7.

[141] Newt. Princ. Lib. II. prop. 40, in schol.

[142] Lib. II. in schol. post prop. 31.

[143] Book I. ch. 2 § 82.

[144] Book I. Ch. 3 § 29.

[145] Ch. 3. of this present book.

[146] Ch. 4.

[147] In Princ. philos. part. 3.

[148] Philos. princ. mathem. Lib. II. prop. 2. & schol.

[149] Ibid. prop 53.

[150] Philos. princ. prop. 52. coroll. 4.

[151] Ibid.

[152] Coroll. 11.

[153] See ibid. schol. post prop. 53.

[154] Princ. philos. pag. 316, 317.

[155] Ch. I. § 7.

[156] Book I. Ch. 3.

[157] Book I. Ch. 3. § 29.

[158] Ibid. Ch. 2. § 30, 17.

[159] Book I. Ch. 3.

[160] Ch. 1. § 7.

[161] Chap. 5. § 8.

[162] Princ. pag. 60.

[163] Street, in Astron. Carolin.

[164] See Chap. 5. §9, &c.

[165] In the foregoing page.

[166] See Newton. Princ. Lib. III. prop. 13.

[167] Chap. 5. § 10.

[168] Princ. Lib. I. prop. 60.

[169] Book I, Chap. 2. § 80.

[170] Princ. philos. Lib. I. prop. 58. coroll. 3.

[171] Newt. Optics. pag. 378.

[172] Newton. Princ. Lib. III. prop. 1.

[173] Newton, Princ. Lib. III. pag. 390,391. compared with pag. 393.

[174] Book I. Ch. 3. § 29.

[175] Princ. philos. Lib. I. prop. 4.

[176] Ibid. coroll.

[177] Newt. Princ. philos. Lib. III. pag. 390.

[178] Newt. Princ. philos. Lib. III. pag. 391, 392.

[179] Book III. Ch. 4.

[180] Newt. Princ. philos. Lib. III. pag. 391.

[181] Ibid. pag. 392.

[182] See Book I. Ch. 2. § 60, 64.

[183] Book I. Ch. 2. § 17.

[184] See Ch. II. § 6.

[185] The second of the laws of motion laid down in Book I. Ch. 1.

[186] Newton. Princ. philos. Lib. III. prop. 6. pag. 401.

[187] Newton’s Princ. philos. Lib. III. prop. 22, 23.

[188] Newton. Princ. Lib. I. prop. 66. coroll. 7.

[189] Menelai Sphaeric. Lib. I. prop. 10.

[190] Vid. Newt. Princ. Lib. I. prop. 66. coroll. 10.

[191] Vid. Newt. Princ. Lib. III prop. 30. p. 440.

[192] Ibid. Lib. I. prop. 66. coroll. 10.

[193] What this proportion is, may be known from Coroll. 2 prop. 44. Lib. I. Princ. philos. Newton.

[194] Princ. Phil. Newt. Lib. I. prop. 45. Coroll. 1.

[195] Pr. Phil. Newt. Lib. I. prop. 66. Coroll. 7.

[196] See § 19 of this chapter.

[197] Phil. Nat. Pr. Math Lib. I. prop. 66. cor. 8.

[198] Ibid. Coroll. 8.

[199] Ibid.

[200] Ibid.

[201] Newt. Princ. Lib. III. prop. 29.

[202] Ibid. prop. 28.

[203] Ibid. prop. 31.

[204] Newt. Princ. pag. 459.

[205] In Princ. philos. part. 3. § 41.

[206] Chap. 1. § 11.

[207] Newton. Princ. philos. Lib. III. Lemm. 4. pag. 478.

[208] Princ. philos. Lib. III. prop. 40.

[209] Book I. chap. 2. § 82.

[210] Princ. philos. Lib. III. pag. 499, 500.

[211] Ibid. pag. 500, and 520, &c.

[212] Princ. Philos. Lib. III. prop. 40.

[213] Ibid. prop. 41.

[214] Ibid. pag. 522.

[215] Ibid. prop. 42.

[216] Newt. Princ. philos. edit. 2. p. 464, 465.

[217] Ibid. edit. 3. p 501, 502.

[218] Ibid. pag. 519.

[219] Ibid. pag. 524.

[220] Newt. Princ. philos. p. 525.

[221] Ibid.

[222] Ibid. pag. 508.

[223] Ibid.

[224] Ibid. pag. 484.

[225] Ibid. pag. 482, 483.

[226] Ibid. pag. 481.

[227] Ibid. pag. 509.

[228] See the fore-cited place.

[229] Ibid. and Cartes. Princ. Phil. part. 3. § 134, &c.

[230] Vid. Phil. Nat. princ. Math. p. 511.

[231] Book I. Ch. 4. § 11.

[232] Ch. 5.

[233] All these arguments are laid down in Philos. Nat. Princ. Lib. III. from p. 509, to 517.

[234] Philos. Nat. Princ. Lib. III. p. 515.

[235] Ch. 5.

[236] See Ch. 1. § 11.

[237] Newt. Princ. Philos. pag. 525, 526. An account of all the stars of both these kinds, which have appeared within the last 150 years may be seen in the Philosophical transactions, vol. 29. numb. 346.

[238] Newt. Princ. Philos. Nat. Lib. III. prop. 6.

[239] Ch. 3. § 6.

[240] Book I. Ch. 2. § 24.

[241] Newt. Princ. Lib. III. prop. 6.

[242] Ch. 3. § 6.

[243] Newt. Princ. philos. Lib. III. prop. 7. cor. 1.

[244] See Book I. Ch. 1. § 15.

[245] Ibid. § 5, 6.

[246] Chap. 2. § 8.

[247] Newt. Princ. Lib. I. prop. 63.

[248] § 8.

[249] See Introd. § 23.

[250] § 4, 5.

[251] Newt. Princ. philos. Lib. I. prop. 74.

[252] Ibid. coroll. 3.

[253] Lib. I. Prop. 75. and Lib. III. prop. 8.

[254] Lib. I. Prop. 76.

[255] Ibid. cor. 5.

[256] Vid. Lib. III. Prop. 7. coroll. 1

[257] Newt. Princ. Lib. III. prop. 8. coroll. 1.

[258] Ibid. coroll. 2.

[259] Book I. Ch. 4. § 2.

[260] Newt. Princ. Lib. III. prop. 8. coroll. 3.

[261] Ibid. coroll. 4.

[262] Book I. Ch. 4.

[263] Lib. II. prop. 20. cor. 2.

[264] Chap. 4. § 17.

[265] Ibid.

[266] Vid. Newt. Princ. Lib. II. prop. 46.

[267] Princ. philos. Lib. II. prop. 49.

[268] Chap. 3. § 18.

[269] Newt. Princ. philos. Lib. I. prop. 66. coroll. 18.

[270] § 8.

[271] Ch. 3. § 5.

[272] Ch. 3 § 17.

[273] Ibid.

[274] See below § 44.

[275] Newton Princ. Lib. III. prop. 19.

[276] Lib. III. prop. 19.

[277] Lib. I. prop. 73.

[278] Lib. III. prop. 20.

[279] Ibid.

[280] Opt. B. I. part. 2. prop. 1.

[281] Newt. Opt. B. 1. part 1. experim. 5.

[282] Ibid. prop. 4.

[283] Newt. Opt. B. 1. part 2. exper. 5.

[284] Ibid exper. 6.

[285] Newton Opt. B. I. prop. 10.

[286] Ibid exp. 9.

[287] Newt. Opt. B. I. part 1. exp 15.

[288] Philos. Transact. N. 88, p. 5099.

[289] Opt B. I. par. 2. exp. 14.

[290] Ibid. exp. 10.

[291] Opt. pag. 122.

[292] Opt. B. I. part 2. exp. 11.

[293] Ibid prop. 4, 6.

[294] Opt. pag. 51.

[295] Opt. Book II. prop. 8.

[296] Opt. Book II. par. 3. prop. 2.

[297] § 17.

[298] Opt. Book II. par. 3. prop. 4.

[299] Opt. Book II. pag. 241.

[300] Ibid. pag. 224.

[301] Ibid. Obs. 17. &c.

[302] Ibid. Obs. 10.

[303] Ibid. pag. 206.

[304] Obser. 21.

[305] Observ. 5. compared with Observ. 10

[306] Ibid. prop. 5.

[307] Observ. 7.

[308] Observ. 9.

[309] Ibid prop. 7.

[310] Opt. pag. 243.

[311] Newt. Opt. B. I. part. 1. prop. I.

[312] Opt. B. I. part. 1. prop. 2.

[313] Opt. B. I. part 1. Expec. 6.

[314] Opt. pag. 67, 68, &c.

[315] Ibid. B. 1. par. 2. prop. 3.

[316] Opt. B. II. par. 3. prop. 10.

[317] Opt. B. II. par. 3. prop. 15.

[318] Ibid. par. 1. observ. 7.

[319] Ibid. Observ. 19.

[320] Opt. B. II. par. 2. pag. 199. &c.

[321] Ibid. par. 4

[322] Ibid. part. 3. prop. 13.

[323] Ibid. prop. 17.

[324] Ibid. prop. 13.

[325] Opt. Qu. 18, &c.

[326] See Concl. S. 2.

[327] B. II. Ch. 1.

[328] Opt. B. III. Obs. 1.

[329] Ibid. Obs. 2.

[330] § 2.

[331] Philos. Trans. No. 378.

[332] § 11.

[333] Philos. Transact No. 375.

[334] Ibid.

[335] Opt. B. II. part 4.

[336] Ch. 3. § 14.

[337] Opt. B. II. part 4. obs. 13.

[338] Opt. pag. 255.

[339] Ch. 3. § 18.