 I've got a small but kind of growing collection of old vintage textbooks, not massive ones, not incredibly rare ones yet, but I collect them mostly to see what the old style of science was like and whether there are any techniques in there that we might have lost or might be interesting to pick up. So I thought let's dive into some and see what they're like. So this one is a refresher course in mathematics by Frederick Cam, FJ Cam. So I did a little bit of digging on this guy and he is the founding editor of a number of kind of hobbyist engineering magazines, including practical mechanics, practical motorists, one of them the practical radio hobbyist, I can't remember the name, it's actually still going. And he's also the younger brother of the guy who designed the Hawker Hurricane fighter plane, so obviously a very engineering focused family. And this one comes from 1943, first published though I think this edition is 1955, so we're talking pre-computer, this is not incredibly old, it's not necessarily going to be completely different maths and it is intended to provide a refresher course for mathematics, for people who have previously mastered the subject but have forgotten the fundamental facts. And it really is just a list of the basics, decimals, how to do division, what all the signs mean, the subjects do get more advanced, we start with the addition and then we're going to plotting graphs. So we know this has not evolved a great deal since the mid 20th century, we would now use computers to do this and get the graphs plotted perfectly but they were always hand drawn back then, you could always find slides on presentations were hand drawn, they had the etched kind of at scale on the slide and doing graphs like that would have been part of that. So let's have a look at what we were talking about, averages, ratios and the one I wanted to look at here we are, the square roots and the cube roots. So this is an algorithm to work out the square root of a number, so there are multiple ways of doing this, there's the Babylonian method, Newton's method, the more generalized version and this algorithm, so if you have been taught how to multiply and divide and add up in school, you were probably taught an algorithm more than anything else, that's what the column multiplication actually is, it is an algorithm, it is a method that you repeat and you should get the right answer, it is not necessarily a mathematically robust explanation of how the system works but it should get you the right answers. So let's say if we can make sense of this, bear in mind that this is a refresher course, it is not meant to teach it, square root of the method of extracting square root is as follows, mark off the number the square root of which is to be found into periods by marking a dot over every second figure starting at the left decimal point, to draw a vertical line next to the left of the figure in a bracket you can kind of get the feeling here that this is really about the notation. Let's see, we can replicate the example for a moment, let's do, what is it, one, one, five, six, off and off every two or bracket, line down, gonna put the answer here, I think that's what it means. Find the largest square in the left hand period and place this root behind the bracket, so that is, okay so we need to find the largest, largest square, implicitly that is square number, so the largest square number, we can see it in the example is nine, so we place the root here and here for some reason, okay, next to square, next to square of this root is subtracted from the person, this is weirdly redundant, the square of the root, so we find the largest square, then you find the root, then you square it, okay, so that's three times three, that's nine, square of this root is subtracted from the first period, the next period is brought down adjacent to the remainder and used as a dividend, so 11 minus nine, that's okay, 11 minus nine, well that didn't help, right, so 11 minus nine, that's two, and then you bring the next period down, that's five, six of 256, so we've got this remaining. Now, multiply the first root find by two and place the pro to the left of the verse line, so three, multiply that by two, place it to the left of the verse line here, so now multiply the root find by two and place this product into the last of this line, then divide it into the left hand figure of this new dividend, ignoring the right hand figure, what does that mean? Now divide it into the left hand figures of the new dividend, ignoring the right hand figure, so that's 6 into 25, 6, 12, 24, okay, that's four, right, so attach the figure obtained, finally subtracting the product, multiply the latest divisor by the figure of the root last of, you just fold it in, four, so 64 times four, multiply, finally subtracting the product from the dividend. You say fold in one more time. It says fold it in. So 64 times, so what's 64, I'm gonna try it 256, subtract that, okay, so we terminated zero, don't we? So what I think this is doing is if you think about what a square number has to be, we have a square x on one side, x on the other, and the total area of that is x squared, so this total area is gonna be 1056, x is gonna be 34, what's happening is we've split this up, I'll not draw it to scale like here, so we have 30 and 4 here, and 30 and 4, so what we've got here is 30 times 30 for 900, so this nine here at the top is not really a nine, it is a nine kind of 100, so we're working out that as a perfect square, and the rest of the square are these three areas which must add up to in total 256, so we're trying to back calculate what that number must be, and if we get it exactly right, then the algorithm terminates, if not we have a little bit of a square left over and we start again on the next period, so we're multiplying it by two because we've got this area here which is 4 times 30, and this area here which is 4 times 30, and there are two of those, so what we've really got is if we ignore that square for one, just ignore that one, we've got two that are going to be four and 30, so if we times that 3 by 2 we get 6 or 60 here, and then we're trying to really back calculate what that number should be, and any remainder is there I think, so I think that's what that's getting at, that's what that's doing, let's try another example that's not in here, let's do 256, I know 256 is 16 squared, that's one positive, so well the answer to this should be 16, let's try this one, so we're going to square root 256, mark off the number of the square root of which is to be found into periods two five six, so we've got one period and one period there, draw a dot over every second figure, draw the vertical line, next find the largest square in the left hand period, well the left hand period is two, the largest square number in that is one, the square of this root is subtracted from the first period, yep that next period is brought down, so two minus one is one, we're in the next one, 156, now multiply the first root found by two, so we'll do that times two and divide it into the left hand figure, so two into 15, one, two, three, four, five, six, seven, seven, two into 15 is seven and a bit, so that's finding us the square root of 256, 17, divide it into the left hand figures of the new dividend, all in the whole ignoring the right hand figure, two into 15, two into 15 is seven, two should be six, next square of this root is subtracted from the first period, the next period is brought, this is your recipe, you fold in the cheese then, 20 sometimes, what's 27, what's seven times seven, 49, 189, 189, so 189, let's find the largest square in the left hand period, two into 15 is seven, I'm gonna have to work a pg of this, let's go, right, I know what's going on here, I have checked, there is a slight ambiguity here, let's start again 256 and why does 256 cause a bit of a problem, so 256, we want to work out that first root it is going to be a one the first figure and we have one two and we're going to do 156. So we've got to divide something into here and it's either going to be and we've got to figure out either 26 times 6 or 27 times 7 because if we put the skesh to 7 here or we put 2 into 15 to get 7 we then multiply 27 by 7 to get 189 that is 26 times 6 is 156 which rounds off perfectly so we would put a 6 here and then we do 26 times 6 256 algorithm terminates reason for marking these off as periods of groups of 2 it's because 10 times 10 is a hundred so that's we are working this out in groups of a hundred because if we have anything left over that becomes another another square that we can deal with we're doing it desk place by place and you can kind of see that that's what's going on when we talk about the screw boot because the key boot is effectively the same algorithm but the periods of placing a dot every third figure so a cube 10 times 10 10 will be a thousand there is a couple of drawbacks to it one as you can see from this example especially the numbers you have to start dividing in by get increasingly big so every time you bring another period down you're adding two figures but it must you're eliminating maybe one and if you get lucky two from the left hand side so this number every time you have to make a new iteration grows one larger and the numbers that you have to then start putting into your head if you want to do this without calculator are much much bigger it would take oh you'd have to then divide here by right you're doing what 12,563 dividing into something or multiplying by something but the advantage of this method is that it is figure by figure each figure is correct we know from the start of this this must have begun with a one that's not going to get better we know that that number is correct and we did it 16 we know that that is correct other methods are iterative so Newton's method for instance of finding roots we want to find a root of a polynomial we kind of guess and we do derivatives to figure it in a polynomial like that x squared minus 2 equals 0 is the equivalent of solving for the square root of 2 but all the numbers in those methods change every time you iterate this method the numbers you just stack on more numbers they are going to be corrected as an algorithm this is how it would have originally been done if you want the numbers to be as precise as possible as quickly as possible that's it here at all some shortcuts so practical men use a practical men there's very very possibly the most engineering words ever written use a vast number of shortcuts in calculations a few of the more useful ones are given to multiply by 5 add not the number to be altered and divide by 2 very straightforward sounds kind of weird but that is the mental shortcut you would do times in by 10 when you're in a decimal base 10 is trivial and then half is also kind of trivial so it's two operations that are kind of trivial and you get a lot of resentment towards things like that when people get confused by new math common core of you in America and the object to it they think oh why aren't you just doing the normal way that I was taught you actually do this in your head and it's more efficient and it's about learning number sense and the ability to do the shortcuts so it's nice to see that this even in the in the 1940s and 50s that was actually being taught I think towards the end we get shapes geometry some mechanics stuff so mechanics are really useful for science not because you will necessarily be using these diagrams and forces and understanding them directly but because what these diagrams are is that they have meaningful forces and directions and masses on them and you build equations from the diagrams and that act of interpreting what an equation means with the diagram and constructing the two out of each other that connection is amazing the useful if you want to do a mathematical science this is this is incredibly practical you want to relate two things together we do see it in physical chemistry we have to relate multiple things together if you want to do chemical kinetics you have to link the you have to link the macroscopic the symbolic and the sub microscopic worlds together it's not too different to how mechanics works you have a symbolic domain and a physical representation of what that means and you have to link them together so that's a really nice thing to see in a book like this and it's nice just to have all of this stuff that is designed to be the refresher of mathematics before you do any science or engineering it's it's interesting how some of its formatted because you see this is pre latex and and pre word and etc so how these were assembled I don't know probably having to typewriter them really difficultly in with a lot of precision instruments and then copying them or will one day a workout so there is nice book to have I might look at some more later