4 Proposition 16: Hence if from the radius with seven cyphers added you subtract its 10000000th part, and from the number thence arising its 10000000th part, and so on, a hundred numbers may very easily be continued geometrically in the proportion subsisting between the radius and the sine less than it by unity, namely between 10,000,000 and 9,999,999; and this series of proportional we name the First table. FIRST TABLE 10,000,000.000 000 0 1.000 000 0 9,999,999.000 000 0 0.999 999 9 9,999,998.000 000 1 0.999 999 8 9,999,997.000 000 3 To be continued up to 9,999,900.000 495 0 Thus from radius, with seven cyphers added for greater accuracy, namely, 10,000,000.000 000 0 subtract 1.000 000 0 you get 9,999,999.000 000 0 from this subtract 0.999 999 9 you get 9,999,998.000 000 1; and proceed in this way... until you create a hundred proportional, the last of which, if you have computed rightly, will be 9,999,900.000 495 0. Proposition 17: The Second table proceeds from radius with six cyphers added, through fifty other numbers decreasing proportionally in the proportion which is easiest, and as near as possible to that subsisting between the first and last numbers of First table. SECOND TABLE 10,000,000.000 000 100.000 000 9,999,900.000 000 99.999 000 9,999,800.001 000 To be continued up to 9,995,001.224 804 Thus the first and last numbers of the First table are 10,000,000.000 000 0 and 9,999,900.000 495 0, in which proportion it is difficult to form fifty proportional numbers. A near and at the same time and easy proposition is 100,000 to 99,999, which may be continued with sufficient exactness by adding six cyphers to radius and continually subtracting from each number its own 100000th part... and this table contains, besides radius which is the first, fifty other proportional numbers, the last of which, if you have not erred, you will find to be 9,995,001.224 804*. Proposition 18: A third table is to be constructed of 69 columns. Each column contains 21 entries constructed much like the first two. The first column begins with five cyphers added, 10,000,000.000 00. Subsequent numbers are found by subtracting the 2000th of the previous entry. Once the last entry has been found it should be recorded then (in subsequent steps) you may disregard any of the least significant digits so your calculations will be easier. Proposition 19: The first numbers of all eh columns must proceed from radius with four cyphers added, in the proportion easiest and nearest to that subsisting between the first and the last numbers of the first column. As the first and the last numbers of the first column are 10,000,000. 000 0 and 9,900,473.578 0, the easiest proportion very near to this is 100 to 99. Accordingly sixty-eight numbers are to be continued from radius in the ration of 100 to 99 by subtracting from each one of them its hundredth part. Proposition 20: In the same proportion a progression is to be made from the second number of the first column through the second numbers in all the columns, and from the third through the third, and from the fourth through the fourth, and from the others respectively through the others. Thus from any number in one column, by subtracting its hundredth part, the number of the same rank in the following column is made, and the numbers should be placed in order as follows. Comum, and the TUTUCTS non e piacca in oruc TORONJ. PROPORTIONALIA TERTIE TABVLA. Secunda Col. 9900000.0000 Prima Columna. 10000000,0000 9995000.0000 9990001.5000 985007.4987 9980014.9950 9895050.0000 9890101.4750 9885157.4137 9880214.8451 9900473.5780 9801468.8413 7nde 4.5. &c. vfq. ad 69 column. &c. vfque ad 5048858.8900 &c. vfque al 5046334.4605 &c. vfque al 504381.2932 &c. vfque ad 5041289.3879 &c. vfque ad 5038768.7435 9703454-1539 vfque tandem ad 4998609.4034 This table, "Proportionals of the Third table," with 69 columns, has as the last number the in the sixty- ninth column being 4,998,609.403 4, roughly half the original number of 10,000,000. *Napier actually had this value incorrect in the original manuscript. Tertia Cel. 9801000.0000 9796099.5000 9791201.4503 9786305.8495 9781412.6967 dendo ad & c.defcen- &c, defcen.co Exercise: Your job is to use modern technology to create all three of these tables. Please label your spreadsheet appropriately with the name of the table (First table, Second table, Third table) appropriately on the tabs. Keep in mind that Napier, and his colleagues, did all these tables and calculations essentially by hand. Their "computing devices" were nothing even close to our modern calculators. Please keep track of how long it takes you to complete this assignment as it will be required on your upload. 4 Proposition 16: Hence if from the radius with seven cyphers added you subtract its 10000000th part, and from the number thence arising its 10000000th part, and so on, a hundred numbers may very easily be continued geometrically in the proportion subsisting between the radius and the sine less than it by unity, namely between 10,000,000 and 9,999,999; and this series of proportional we name the First table. FIRST TABLE 10,000,000.000 000 0 1.000 000 0 9,999,999.000 000 0 0.999 999 9 9,999,998.000 000 1 0.999 999 8 9,999,997.000 000 3 To be continued up to 9,999,900.000 495 0 Thus from radius, with seven cyphers added for greater accuracy, namely, 10,000,000.000 000 0 subtract 1.000 000 0 you get 9,999,999.000 000 0 from this subtract 0.999 999 9 you get 9,999,998.000 000 1; and proceed in this way... until you create a hundred proportional, the last of which, if you have computed rightly, will be 9,999,900.000 495 0. Proposition 17: The Second table proceeds from radius with six cyphers added, through fifty other numbers decreasing proportionally in the proportion which is easiest, and as near as possible to that subsisting between the first and last numbers of First table. SECOND TABLE 10,000,000.000 000 100.000 000 9,999,900.000 000 99.999 000 9,999,800.001 000 To be continued up to 9,995,001.224 804 Thus the first and last numbers of the First table are 10,000,000.000 000 0 and 9,999,900.000 495 0, in which proportion it is difficult to form fifty proportional numbers. A near and at the same time and easy proposition is 100,000 to 99,999, which may be continued with sufficient exactness by adding six cyphers to radius and continually subtracting from each number its own 100000th part... and this table contains, besides radius which is the first, fifty other proportional numbers, the last of which, if you have not erred, you will find to be 9,995,001.224 804*. Proposition 18: A third table is to be constructed of 69 columns. Each column contains 21 entries constructed much like the first two. The first column begins with five cyphers added, 10,000,000.000 00. Subsequent numbers are found by subtracting the 2000th of the previous entry. Once the last entry has been found it should be recorded then (in subsequent steps) you may disregard any of the least significant digits so your calculations will be easier. Proposition 19: The first numbers of all eh columns must proceed from radius with four cyphers added, in the proportion easiest and nearest to that subsisting between the first and the last numbers of the first column. As the first and the last numbers of the first column are 10,000,000. 000 0 and 9,900,473.578 0, the easiest proportion very near to this is 100 to 99. Accordingly sixty-eight numbers are to be continued from radius in the ration of 100 to 99 by subtracting from each one of them its hundredth part. Proposition 20: In the same proportion a progression is to be made from the second number of the first column through the second numbers in all the columns, and from the third through the third, and from the fourth through the fourth, and from the others respectively through the others. Thus from any number in one column, by subtracting its hundredth part, the number of the same rank in the following column is made, and the numbers should be placed in order as follows. Comum, and the TUTUCTS non e piacca in oruc TORONJ. PROPORTIONALIA TERTIE TABVLA. Secunda Col. 9900000.0000 Prima Columna. 10000000,0000 9995000.0000 9990001.5000 985007.4987 9980014.9950 9895050.0000 9890101.4750 9885157.4137 9880214.8451 9900473.5780 9801468.8413 7nde 4.5. &c. vfq. ad 69 column. &c. vfque ad 5048858.8900 &c. vfque al 5046334.4605 &c. vfque al 504381.2932 &c. vfque ad 5041289.3879 &c. vfque ad 5038768.7435 9703454-1539 vfque tandem ad 4998609.4034 This table, "Proportionals of the Third table," with 69 columns, has as the last number the in the sixty- ninth column being 4,998,609.403 4, roughly half the original number of 10,000,000. *Napier actually had this value incorrect in the original manuscript. Tertia Cel. 9801000.0000 9796099.5000 9791201.4503 9786305.8495 9781412.6967 dendo ad & c.defcen- &c, defcen.co Exercise: Your job is to use modern technology to create all three of these tables. Please label your spreadsheet appropriately with the name of the table (First table, Second table, Third table) appropriately on the tabs. Keep in mind that Napier, and his colleagues, did all these tables and calculations essentially by hand. Their "computing devices" were nothing even close to our modern calculators. Please keep track of how long it takes you to complete this assignment as it will be required on your upload