Mathesis enucleata, or, The elements of the mathematicks by J. Christ. Sturmius ; made English by J.R. and R.S.S.

SCHOLIUM.

HEre we will shew the Excellent use of these last Proposition•• in making the Tables of Signs. For having found above, supposing the Radius of 10000000 parts, the sides of the chief Regular Figures, if they are Bisected, you will have so many Primary Sines; viz. from the side of the Triangle the side of 60 Degrees 8660754, from the Side of the Square, the Sine of 45° 7071068; from the Side of the Pentagon, the Sine of 36° 5877853; from the Side of the Hexa∣gon, the Sine of 30° 5000000; from the Side of the Octagon, the Sine of 22° 30 3826843; from the Side

of the Decagon the sine of 18°=3090170; from the side of the quindecagon lastly the sine of 12°=2079117. From these seven primary sines you may find afterwards the rest, and consequently all the Tangents and Secants according to the Rule we have deduc'd, n. 3. Schol. 5. Prop. 34. and which Ph. Lansbergius illustrates in a prolix Example in his Geom. of Triangles Lib. 2. p. 7. and the following. But af∣ter what way, having found these greater numbers of sines, Tangents, &c. Logarithms have been of late accommodated to them, remains now to be shewn, which in brief is thus; viz. the Logarithms of sines, &c. might immediately be had from the Logarithms of vulgar numbers, if the tables of vul∣gar numbers were extended so far, as to contain such large numbers; and thus the sine e. g. of o gr. 34. which is 98900 the Logarithm in the Chiliads of Vlacquus is 49951962916. But because the other sines which are greater than this are not to be found among vulgar numbers (for they ascend not be∣yond 100000, others only reaching to 10000 or 20000) there is a way found of finding the Logarithms of greater num∣bers, than what are contained in the Tables. E g. If the Logarithm of the sine of 45° which is 7071068 is to be found, now this whole number is not to be found in any vulgar Ta∣bles, yet its first four notes 7071 are to be found in the vul∣gar Tables of Strauchius with the correspondent Logarithm 3. 8494808, and the five first 70710 in the Tables of Vlac∣quus with the Log. 4. 8494808372. One of these Loga∣rithms, e g. the latter, is taken out, only by augmenting the Characteristick with so many units, as there remain notes out of the number proposed, which are not found in the Tables, so that the Log. taken thus out will be 6. 8494808372. Then multiply the remaining notes of the proposed number by the difference of this Logarithm from the next following, (which for that purpose is every where added in the Vlacquian Chiliads, and is in this case 61419) and from the Product 4176492 cast away as many notes as adhere to the propo∣sed number beyond the tabular ones, in this case 2; for of the remainder 41764, if they are added to the Logarithm before taken out, there will come the Logarithm requi∣red 6. 8494850136, viz. according to the Tables of Vlac∣q••us, wherein for the Log of 10 you have 10,000,000,

000; but according to those of Strauchius which have for the Logarithm of 10 only 10,000,000, you must cut off the three last notes, that the Logarithm of the given sine may be 6.8494850; as is found in the Strauchian and other tables of sines, except that instead of the Characteristick 6 there precedes the Characteristick 9, whereof we will add this reason: If the Characteristicks had been kept, as they were found by the rule just now given, the Logarithm of the whole sine (which is in the Strauchian Tables 10,000,000) would have come out 70,000,000, incongruous enough in Trigo∣nometrical Operations. Wherefore that Log. of the whole sine might begin from 1, for the easiness of Multiplication and Division they have assumed 100,000,000; the Cha∣racteristick being augmented by three, wherewith it was con∣sequently necessary to augment also all the antecedent ones; and hence e. g. the Logarithm of the least sine 2909 begins from the Characteristick 6, which otherwise according to the Tables of vulgar numbers would have been 3.

Having found after this way the Logarithms of all the sines (altho' here also if you have found the Logarithms of the signs of 45° and moreover the Logarithm of 30, the Loga∣rithms of all the rest may be compendiously found by addition and substraction from a new principle which now we shal omit) the Logarithms of the Tangents and Secants may easi∣ly be found also, only by working, but now Logarithmically, according to the Rules of Schol. 5. Prop. 34. n. 5. and 6.