ENPages
Page 7
Lecture 1st All the soil is composed of the particles of rocks that have been washed off. A grain of gold is about as large as the 150th part of a 5 cent piece, and it can be hammered out untill the 2,000,000 part can be seen by the naked eye. The sun is about 880,000 miles in diameter. The earth turns round at the rate of 19 miles per second Ropes are shortened by welling owing to the capilary attraction.
To tell whether stone will stand frost for building take sulphate of soda or Glanbers salts and make a strong solution and soak the stone in it then place it in the ovn and the water will exaporate and the salts chrystallize and crack the stone if it is not a good material for building.
Lecture II Lecture 2
Page 9
Demonstrations of the Propositions of the Second Book of Daries Geometry
Defonition 1st Ratio is the mutual relation between two quantities of the same kind and is expressed by the quotient arriving by dividing the second by the first. Thus the Ratio of A to B is B/A of M to N is N/M etc.
Def 2nd Proportion is an equatity of ratios
3rd When four quantities are so related that the second divided by the first is equal to the 4th divided by the 3st they are said to be in proprtion; that is the 1st is said to have to the second the same ratio that the 3rd has to the 4th or the 1st is to the 2nd as the 3rd is to the 4th, Thus if B/A is equal to D/C then A:B::C:D and if N/M = Q/P then M:N::P :Q.
4th When four quantities are in proportion the 2nd divided by the 1st is equal to the 4th divided by the 3rd, Thus if A:B::C:D then B/A = D/C and if M:N::P:Q then N/M = Q/P.
Page 10
Proposition 1st Theorem When four quantities are in proportion the product of the two extremes is equal to the product of the means, We have by hypothesis M:N::P:Q hence by Def 4th N/M = Q/P then learning it of fractions we have N x P = M x Q, Cor, We have by hypothesis M:N::N:Q then by Def 4th N/M = Q/N learning of fractions we gives N^2 = Q x M Q, E, D
Propositions 2nd Theorem If the product of the quantities be equal to the product of two other quantities two of them will be the extremes and the other two the means of a proportion. We have by hypothesis M x Q = N x P 1st divide each into N x Q and we have N/M = Q/P Hency M is to N as P is to Q an P:Q::M:N. 2nd divide each by N x Q and we have M/N = P/Q hense N:M::Q:P or Q:P::N:M.
3rd divide each into P x Q and we have P/M = Q/N hense M:P::N:Q or N:Q::M:P, 4th divide each by P x Q and we have M/P = N/Q hense P:M::Q:N or Q:N::P:M Q, E, D,
Scholium, It is evident that we can take either factor as the 1st extreme and then the factor associated with it will be the other extreme, It is also evident that the other factors may be taken in any order we please,