M. Blundevile his exercises containing sixe treatises, the titles wherof are set down in the next printed page: which treatises are verie necessarie to be read and learned of all yoong gentlemen that haue not bene exercised in such disciplines, and yet are desirous to haue knowledge as well in cosmographie, astronomie, and geographie, as also in the arte of navigation ... To the furtherance of which arte of navigation, the said M. Blundevile speciallie wrote the said treatises and of meere good will doth dedicate the same to all the young gentlemen of this realme.

Of proportion of the greater inequalitie there be two kindes, Simplex and Multiplex, that is to say, simple and manifold.

SImplex, is when the Antecedent, that is to say the former num∣ber containeth the consequent, that is to say the latter number once and somewhat more, which ouerplus must alwaies bee lesse then the consequent it selfe, as 5. containeth 4. once and one parte thereof more, for if you diuide 5. by 4. the quotient will be 1. and ¼ ouer. Againe, this proportion is twofold, that is Superparticu∣ler, and Superpartient: Superparticular, is when the Antece∣dent containeth the consequent once and some one part thereof, as 3. to 2. for 3. containeth 2. once and one halfe thereof, which is called Sesquialtera, or as 4. to 3. for 4. containeth 3. once and one third part thereof, and is called Sesquitertia, the like is to be said of Sesquiquarta, Sesquiquinta, Sesquisexta, and so forth in∣finitely as this Table sheweth.

Super∣particu∣lar pro∣portiōs are these & such like.	Sesquialtera as 3. to 2. 6. to 4. 9. to 6.	which is as much as.	1 ½
Sesquitertia as 4. to 3. 8. to 6. 12. to 9.	1 ⅓
Sesquiquarta as 5. to 4. 10. to 8. 15. to 12.	1 ¼
Sesquiquinta as 6. to 5. 12. to 10. 18. to 15.	1 ⅕
Sesquisexta as 7. to 8. 14. to 12. 21. to 18.	1 ⅙
Sesquiseptima as 8. to 7. 16. to 14. 24. to 21.	1 1/7
Sesquioctaua as 9. to 8. 18. to 16. 27. to 24.	1 ⅛
Sesquinona as 10. to 9. 20. to 18. 30. to 27.	1 1/9
Sesquidecima as 11. to 10. 22. to 20. 33. to 30	1 1/10
Sesquiūdecima as 12. to 11. 24. to 22. 36. to 33	1 1/11
Sesqiuduodec. as 13. to 12. 26. to 24. 39. to 36	1 1/12

But superpartient is when the Antecedent containeth the con∣sequent once and some parts thereof, that is to say more parts then one, as 5. to 3. for 5. containeth 3. once and 2. third parts thereof, which is called Superbipartiens tertias, of which kinds are these set downe in the Table following.

Proportions superpartient are these and such like.	Superbipar¦tiens.	Tertias as 5. to 3. 10. to 6. 15. to 9.	which is as much as.	1 ⅔
Quintas as 7. to 5. 14. to 10. 21. to 15.	2 ⅖
Septimas as 9. to 7. 18. to 14. 27. to 21.	1 2/7
Nonas as 11. to 9. 22. to 18. 33. to 27.	1 2/9
Vndecimas as 13. to 11. 26. to 22. 39. to 33.	1 2/11
Decimas tertias as 15. to 13. 30. to 26. 45. to 39.	1 2/13
Supertri∣partiens.	Quartas as 7. to 4. 14. to 8. 21. to 12.	which is as much as.	1 ¾
Quintas as 8. to 5. 16. to 10. 24. to 15.	1 ⅗
Septimas as 10. to 7. 20. to 14. 30. to 21.	1 3/7
Octauas as 11. to 8. 22. to 16. 33. to 24.	1 ⅜
Decimas as 13. to 10. 26, to 20. 39. to 30.	1 3/10
Vndecimas as 14. to 11. 28. to 22. 42. to 33.	1 3/11
Superqua∣dripartiens.	Quintas as 9. to 5. 18. to 10. 27. to 15.	which is as much as.	1 ⅘
Septimas as 11. to 7. 22. to 14. 33. to 21.	1 4/7
Nonas as 13. to 9. 26. to 18. 49. to 27.	1 4/9
Vndecimas as 15. to 11. 30. to 22. 45. to 33.	1 4/11
Decimas tertias as 17. to 13. 34. to 26. 51. to 39.	1 4/13
Decim quintas as 19. to 15. 38. to 30. 57. to 45.	1 4/15
Superquin∣tupartiens.	Sextas as 11. to 6. 22. to 12. 33. to 18.	which is as much as.	1 ⅚
Septimas as 12. to 7. 24. to 14. 36. to 21.	1 5/7
Octauas as 13. to 8. 26. to 16. 39. to 24.	1 ⅝
Nonas as 14. to 9. 28. to 18. 42. to 27.	1 5/9
Vndecimas as 16. to 11. 32. to 22. 48. to 33.	1 5/11
Duodecimas as 17. to 12. 34. to 24. 51. to 36.	1 5/12
Supersextu∣partiens.	Seqtimas as 13. to 7. 26. to 14. 39. to 21.	which is as much as.	1 6/7
Vndecimas as 17. to 11. 34. to 22. 51. to 33.	1 6/11
Decimas tertias as 19. to 13. 38. to 26. 57. to 39.	1 6/13
Decimas septimas as 23. to 17. 46. to 34. 60. to 51.	1 6/17
Decimas nonas as 25. to 19. 50. to 38. 75. to 57.	1 6/19
Vicessimas tertias as 29. to 23. 58. to 46.	1 6/23

Hitherto of Simplex proportio. Now of Multiplex proportio.

Multiplex proportio is when the Antecedent containeth the consequent more then once, as 6. to 2. for 6. containeth 2. thrée tunes, which is called Tripla proportio, Also 12. to 5. for 12. comprehendeth 5. twise and ⅖. And this Multiplex proportio is twofolde, that is either exact or not exact.

Multiplex exact, is when the Antecedent containeth the conse∣quet more then once, and nothing remaineth, as 4. to 2. 6. to 3. &c. whereof are infinite kinds, as Dupla, Tripla, and so forth as this Table sheweth.

The kinds of Multi∣plex exact are these and such like.	Dupla as 4. to 2. 6. to 3. 8. to 4.	which is as much as.	2/1
Tripla as 6. to 2. 9. to 3. 12. to 4.	3/1
Quadrupla as 8. to 2. 12. to 3. 16. to 4.	4/1
Quintupla as 10. to 2. 15. to 3. 20. to 4.	5/1
Sextupla as 12. to 2. 18. to 3. 24. to 4.	6/1
Septupla as 14. to 2. 21. to 3. 56. to 8.	7/1
Octupla as 16. to 2. 24. to 3. 32. to 4.	8/1
Nondupla as 18. to 2. 27. to 3. 36. to 4.	9/1
Decupla as 20. to 2. 30. to 3. 40. to 4.	10/1
Vndecupla as 22. to 2. 33. to 3. 495. to 45.	11/1

But Multiplex not exact, is when the Antecedent containeth the consequent more then once and some thing remaineth ouer, as 5. to 2. for 5. containeth. 2. twise and one remayneth, and this is also twofolde, that is, Multiplex superparticularis, and Mul∣tiplex superpartiens

Multiplex superparticularis, is when the Antecedent contai∣neth the consequent more then once & one only remaineth, as 7. to 3. for 7. containeth 3. twise and one only remaineth, wherof are diuers kinds, as Duplex sesquialtera, Duplex sesquitertia, Tri∣plex sesquisexta, & so forth as y^e table hereafter following sheweth.

But Multiplex superpartiens, is when the Antecedent contai∣neth the consequent more then once & the remainder is more then 1. as 8. to 3. for 8. containeth 3. twise and 2 thirds ouer, whereof there be many, kinds, as Dupla superbipartiens tertias, Dupla supertripartiens quartas, and so forth as this Table following sheweth, which comprehendeth both kinds, that is Multiplex superparticularis, and Multiplex superpartiens

Multiplex not exact, is either

• Multi∣plex Su∣perparti∣cularis or els.
  • Duplex.
    • Sesquialtera as 5. to 2. 15. to 6.
    • Sesquitertia as 7. to 3. 21. to 9.
    • Sesquiquinta as 11. to 5, 22. to 10.
  • Triplex.
    • Sesquisexta as 19. to 6. 38. to 12.
    • Sesquiseptima as 22. to 7. 44. to 14.
    • Sesquioctaua as 25. to 8. 50. to 16.
  • Quadru∣plex.
    • Sesquinona as 37. to 9. 148. to 36.
    • Sesquidecima as 41. to 10. 82. to 20.
    • Sesquiundecima as 45. to 11. 90. to 22.
  which is.
  • 2 ½
  • 2 ⅓
  • 2 ⅕
  • 3 ⅙
  • 3 1/7
  • 3 ⅛
  • 4 1/9
  • 4 1/10
  • 4 1/11
• Multi∣plex Su∣perparti∣ens.
  • Dupla.
    • Superbipar¦tiens.
      • Tertias as 8. to 3.
      • Quintas as 12. to 5.
    • Supertri∣partiens.
      • Quartas as 11. to 4.
      • Quintas as 13. to 5.
    • Superqua∣dripartiens.
      • Quintas as 14. to 5.
      • Septimas as 18. to 7.
  • Tripla.
    • Superquin∣tupartiens.
      • Octauas as 29. to 8.
      • Nonas as 34. to 9.
    • Supersextu∣partiens.
      • Vndecimas as 39. to 11.
      • Decim, tertias as 45. to 13
    • Superseptu∣partiens.
      • Octauas as 31. to 8.
      • Nonas as 34. to 9.
  • Quadru∣pla.
    • superbi∣partiens.
      • Tertias as 14. to 3.
      • Quintas as 22. to 5.
    • Superqua∣dripartiens.
      • Quintas as 24. to 5.
      • Nonas as 40. to 9.
    • Superquin∣tupartiens.
      • Sextas as 29. to 6.
      • Septimas as 33. to 7.
  which is.
  • 2 ⅔
  • 2 ⅖
  • 2 ¾
  • 2 ⅗
  • 2 ⅘
  • 2 4/7
  • 3 ⅝
  • 3 5/9
  • 3 6/11
  • 3 6/13
  • 3 ⅞
  • 3 7/9
  • 4 ⅔
  • 4 ⅖
  • 4 ⅘
  • 4 4/9
  • 4 ⅚
  • 4 5/7

Thus much of proportion of the greater inequalitie: Now wee will speake somewhat of proportion of the lesser in∣equalitie.