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CPSC 409 Sept. 12

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New materials. Start the recording. The

ginter quadrant. So these are all interrelated.

interrelated.

They were used for the same task, but

they weren't some of the same task, but

they weren't identical. The astrolab

Help you remember which is which. You

should be able to recognize the devices.

If I show you device, you should be able

to recognize it. Um, say on an exam,

this was circular quad. Where do you get

the quad? Quad is four.

Buy is two, try is three, quad is four.

So the quad comes from essentially one

quarter of it. You have four quadrants

inside that circle. Helps you remember

these ones. Um the winter's quadrant.

There were other quadrants, but the

glutter's quadrant is the one that we

talk about. There's a labor's quadrant

for instance, but I won't talk about

that similar. Um, but I at least

mentioned that there was other quadrants

because Gunter didn't come up with it.

Gunter's quadrant was probably the most

wellknown. It was um quite well used. I

focused on just one based on the aster

circular chopped off oops chopped off a

quarter out of it. Um,

solve the same sort of problems. We

could determine for instance by pointing

to the north star

depending on this string where it's

hanging down. There's a little weight

here. our latitude or potentially we

could determine time based on pointing

at the sun. If we know our location the

sun should be and we know what what day

of the year the date it is based on say

position of the sun. We can then

determine time. You say well use time

pieces. Well, they didn't have have

electronics let alone mobile phones let

alone even mechanical clocks. It's

actually harder. If you see the old

mechanical clocks, if you even know what

I'm talking about, something like Big

Ben, a big mechanical clock, supposed

like I don't have a wall clock here, a

wall clock. It's actually very hard to

develop machining for those ones to

produce those ones because the intricate

gear work and other things and precision required.

required.

So these devices, you think what what's

the value of determining location time?

That's how they did it because the

modern-day devices were we weren't even

close to being able to develop these

things. Again, necessity is the mother

invention and they really needed these

tools for various tasks. Turning the

physically the sector is closest to Oh,

I skipped a quadrant video here. So,

show a bit of the the quadrant video in class.

class.

Must have pressed too quickly. And I

have that queued up just to Oops. I

actually have something for discussion

later. And I decided to use Cold Pilot

to actually help out on that one.

It gave a pretty good answer

explanation of how these things can be

used. Very useful. Actually forgotten

that I knew that I knew this, but I

forgot how it worked. I took a um part

of survival training out in the

wilderness. It was part of some other

other stuff education I had. Um saying

that the military people have to survive

out out in the wilderness to be able to

navigate. Then you say, "Well, use your

GPS doesn't always work." when they show

cases of the military, US soldiers out

in the middle of nowhere. They showed

one case with the soldiers in Iraq

navigating and um getting lost. That

actually was a true story in Nazaria and

the US soldiers got captured in Iraq. Um

part of a convoy went off off yonder.

They weren't kidding. So any it still

had some uses even the modern times. You

can watch the rest of it cuz I have the

link in here. But just to give you some

idea of how these devices can be used

have two parts. I clicked on the wrong

one. Easy misclick

the sector. So I put this these images here.

here.

This is the full circle for the

astrolab. This is the quadrant one

quarter of it. The sector and see

visually is much like the quadrant

except this is a solid piece. This was

two pieces. And there's a reason because

this one you cannot stick down the bore

of a cannon and it was actually used for that.

that.

So who came up with the sector with the sector?

Galileo did not come up with this. Many

other people um were credited with

writing about the scription. They had

their own device, how it works, how how

you can build it.

Uh so it goes back to roughly in this

period when there was many people who

were publishing. 1584 Galileo

early writings indicate that he was

working on it and then he actually

published something formally 16006.

That tells you he wasn't the first but

his was most widely distributed. So

people tended to associate the sector

the most with Galileo. So I'm trying to

attribute um proper credit. Galileo

certainly was most wellknown and

associated with it but he wasn't the one

who came up with it. Um with the sector

you can see I won't go through the

details but you can see it's very

similar. You have that string that you

have here and you have that string you

have here weighted string. So you can

you can determine some use it for some

of the same tasks that you can you can

use a sector for some of the same tasks.

So each one of uses the bars here that

you can use for a quadrant. Each one of

these is marked out. So you could

actually determine an angle for the

artillery piece, the cannon. So you know

with respect to the ground, this is 90°

a right angle here. And of course you

wouldn't be firing your cannon like

this. Usually it goes outside, but

apparently they had to tell some of the

militaries, don't do that because then

it' come down and people would get

injured by the bullet.

Okay. So you probably wouldn't fire this

way unless you're trying to destroy your

own position so you're not being

captured. And then you can have marked

off with this

with this string here

angles. So if it was pulled up all the

way here, you'd have a marking for 90°.

As you turn it down here, that string,

you'd have a marking here

for the 80° and mark off for 70° and so

on. So you know, and they use these

calculations. However, I should actually

put this in context. Why? Because if

we're trying to hit someone, we're not

just shooting for shooting sake, but

we're trying to have this angle. Say

it's a about 45° given about this

distance to do the calculations so that

it should should hit the enemy that

enemy fortification. This is the easy

case to imagine a fixed one that doesn't

move a castle over here. But it wasn't

such an easy sort of thing. Um you can't

just take a ruler and measure this out.

So you kind of guesstimate this. use

some calculating device to be able to

figure out okay um given from here to

here and various other inputs besides

just the angle what angle will probably

reach over there and it was a fairly

complicated thing um it was a problem to

be solved now has anyone ever heard that

term hand show hands here kelp no

we need to get one warship is firing

another and he talk about a firing

solution solution. Or they might say if

you're in a a fighter craft, engage the

other fightercraft, but for God's sake,

don't get in the firing solution of our

own uh carrier ship. Anyone heard that

term from a movie or book, the firing

Jeffrey. Okay, pe of your head. It was

such a tough problem. It was a problem.

You needed to find solution. Find that

solution was not easy. If you want to

see well, you can actually look it up

and see if what videos will have it. If

you're lucky, you'll see the the pop

culture use. Maybe also see how it was

calculated. Some historical type videos.

The most modern one that I can think of

is even in science fiction the reboot of

Battlestar Galactica. Apollo the the the

wing leader saying, "For God's sakes,

don't get galactic as their mother ship

sparring solution. They're engaging with

an enemy ship." And it was a tough

calculation. Now each of these points

was marked off as I said here. However,

I actually did a little bit of checking

here. First I used co-pilot I admit AI

program. However, then I checked it

sources because you don't just rely on

this because it pulls it off the web and

it got it off two good sources. One was

scientific American but another one was

Oxford Scientific something their

publication. It was independent of

Oxford but it's tied with Oxford. So it

was correct. And I had a hunch about

this and this is the first semester I

actually confirmed this. What I'm

getting at is why did this check? The

angles were all marked off

in the sector. So you had 90°, however

it worked, 80 and so on. But the 0

degrees, that's this one right here,

was was was blank. They didn't have a

mark. And if here's another term that

comes from military. You've probably

heard this one more likely than fire solution.

solution.

Don't immediately shoot

until you see the whites of their eyes.

That's one of them. That George

Washington, the American Revolution.

You're so close you can see. Or they

might say

wait until they're at point blank range.

That's an actual term. And probably I

think a number of you have heard the

fire at point blank range. Where does

that come from? It comes exactly from

this. This I knew this Mike Williams

talks about in this course. And they

left a blank for where the zero degree

was. Not zero degree is here where it's

on elevated say 5 10 and so on. But zero

because zero was viewed with suspicion.

So they didn't want to mark it off. And

myself, my own historical background, I

took well I've read a lot of history.

I'm not a history scholar by any means.

Um but

I know that historically there was um

zero in

some cultures including the Christian

cultures was regarded with suspicion and

I I knew that but I also knew that from

the history courses magic science

history magic science and religion that

in Christian cultures in particular a

vacuum was antitheical

antitheical

or had with the Christian belief. I

forgot why. Actually, I had to look that

one up. But I had a hunch that these two

things were tied together. I did a did a

this is where I saw Scientific Americans

article as well as the one from Oxford

University, Oxford something. It was it

wasn't Oxford University Press but was

tied to Oxford where they said a vacuum

is viewed as being um

bad in terms of Christianity because the

the the Christian belief was God is

everywhere. But if we have nothing then

that's got to be satanic. So it does

actually tie together why zero was

viewed with suspicion because a vacuum

is viewed as being anti-Christian and

because zero is viewed with suspicion

with the Christian view and not just

Christian before pre-Christian they left

it marked off how do you count out say

uh a negative value someone asked can we

actually do with the finger reckoning a

multiplication with remember it was only

five onwards but less than five you have

negative numbers you can't represent a

negative quantity by with physical

objects representing nothing. Um the

ancient Greeks regarded this as not

mathematics but actually philosophical.

Now what you don't need what you do need

to know is differences between these two

and you probably should know the you

should know the operations of the

sector. It's marked off at various

points and it doesn't hurt to know. You

should also know the the um the the

firing solution, not so much

terminology, but as an example, the

application as well as the application

of the fire at point blank range. What

you don't need to know, however, is the

um is the other parts that that strained

other history courses, the religion for

instance, and philosophy. Giving you

some context, I was actually not as

pleased. My hypothesis, my hunch was

right that zero is viewed suspicion in

Europe at least because God is

everywhere and a vacuum is viewed with suspicion.

suspicion.

So I typed that in some form into

co-pilot the AI with bin but I viewed it

source where it was basing its answer on

and then I looked at the article very

quickly before class and the sources

were good sources

questions here.

So now I can impress your friends with

these your geekier or your historical

friends, historic-minded friends, where

these terms come from, especially the

fire at point blank range because it

wasn't marked off like the other ones.

Uh Har, what's your question? You have

Uh yeah, hi. Uh, I heard that point

blank comes from like French where it

was like a term for archery that the

target had like a white in the center

and if you hit the center that's like

point blank range. It's so hard you'd

have to be very close to it.

Well, from what I unless it's change of

archery, the actual bullseye is a black

one. Might be a specific difference. It

might be an additional one. But this

point blank definitely comes from here.

That's from Mike Williams actually

rather than a web search. So, point

blank. Um,

perhaps they're considering it's another

tie because you're aiming at short range

here. But point blank the words maybe

it's a I don't know French wall. Not the

point tying into a French word. What

point might be, but it definitely does.

The artillery comes from the artillery.

Maybe it's an alternative one. Common

And it makes more sense

unless the targets were a little

different. But even the target shooting,

every time you now think about it, every

time they depict a target in medieval

times, it was still the black one

as opposed to the white one. And the

white one, you kind I'm just putting

some thought in here. The only way a

white one would work if you had a black

background cuz it would be next to

invisible to see. Just thinking in terms

of practical cuz you're shooting from

very far away. Unless you had a contrast

of white, you couldn't really hit it.

It's hard enough cuz it's small and you

don't want to make it invisible either.

So that one I'd say I haven't verified

but this one I do know for sure.

However, to get that firing solution,

the computations I'm not I even gave you

a um

a foreshadowing of this. The angle is

not the only factor that goes into be

able to determine basically the the only

calculation necessary to be able to hit

your target. There are other things what

they call the bore or the basically the

size of the weapon you're using. bors,

lacrosse. Um, to give you some idea, the

guns on the warships, and it's easiest

to see with the older ones. In World War

II, they had small guns that were like heavy

heavy

heavy caliber machine guns. In fact,

they had some machine guns or aerial

cannons. The bullets are about so big

and it was about this this about this

this big, this long, and about this

wide. Those ones had a very short range

of machine guns. Smaller bore ones. They

didn't have as much thrust. They were

meant to perhaps anti-personnel or for

hitting airplanes that weren't very far

away. Uh as they got to larger and

larger bores, the most you don't need to

know the specifics of this for example,

but you know bore is that this was one

of the things that was in the calculations.

calculations.

That's what you need to know. But the

the bore with the World War II examples

you'll need to know. The longest range

um artillery piece, naval artillery gun

that they had was about 25 miles. That

was for the ones that were 16 in plus or

minus a little bit across. And the

shells on those battleships were about 6

ft high and they could propel those

shells 25 miles. So the size of the gun

was one of the factors that affected it.

The X in this case, it's an overloaded

um symbol. It means different things,

but in this case it means gunpowder. The

amount of charge that you use the

gunpowder. If you use more, it has more

propellant and it shoots it further.

Intuition over here. What this is

illustrating is what you're shooting.

This is my depiction of a cannonball, a

solid metal ball. There's a reflection.

This, on the other hand, is kind of

roughly the equivalent, a cannon

equivalent of a shotgun propellant. They

called it grapeshot at least some of the

times it this one would shoot one one

big solid metal piece. This one would f

was essentially a whole bunch of little

pieces and the ranges would differ

between these two

and both had their purposes. This is

obviously for salt imp placements. If

you're hitting a wall or when they had

um military vehicles a tank it wasn't a

cannon but it was a shell you're trying

to damage that. On the other hand, these

ones, the shotgun type approach were

anti-personnel. Yes, the cannon could do

devastating damage to a person,

artillery, a infantry bed and the guy

behind them, but they tend to scat. They

tend to spread out. This one would shoot

out in the cone and hit many of the um

people. Now again, you don't need to

know that part, but the distance that

this goes versus this is different. It

will differ. So these things, the type

of um ammunition, if you will, that was

being used was part of the firing

solution. The amount of gunpowder was

part of the firing solution. This one's

an easy one to see, as well as the size

of the gun. So you should know that

these were some of the factors in terms

of the exam that were involved in

determining basically do we hit our

target. And I've simplified it. This I

said the easiest case to imagine is if

you have a fixed imp placement it's not

moving in other words or you have the

enemy who's just sitting there for some

reason they're not attacking not retreating

retreating

that distance you're estimating is fixed

but the enemy if the enemy is attacking

coming towards you and you're trying to

hit them whether someone said with the

archery or we're trying to shoot them

with artillery or even before that have

um the catapults were launching the

catapults the angle was a necessary part

of the calculation be hit them when we

actually when you got to naval warfare

it was even harder because we had moving

ships you're moving they're moving and

we had other factors so the calculations

the calculations still have to be done

with modern weapons a missile but it's

automatically done with electronic

devices in these cases they use various

things like lookup tables to um find a solution.

So it definitely had real world

applications. The proportional compass

basically the quick answer for one use

of this proportional compass where and

how you would use it the application.

Say you they didn't have tools to do

this. You had a circle of this size that

was already d drawn and you wanted this circle

circle

um this perfect circle here but in a

different proportion twice as large.

Some of their mathematical calculations

to solve it was determined by basically

the area of something. Well, you had to

be accurate. Say we needed a circle that

was double or triple or half the size.

How do we actually retain that perfect

circle? they would use with the

proportional compass. You see it's got

this ability to move forwards and

backwards and what it could do is if

it's say in the middle we have perfect

proportion. We reduplicate that circle.

We have this pointed towards the circle

it's already drawn on the piece of paper

and it wasn't easy to draw it the first

time. And then this one would be so have

it that have it with one. We could have

it with another. Maybe you have this

paper up here to draw one down here.

You're trying to draw some pens. And

then he basically traces around and you

have if it's right in the middle a

circle a perfect circle that is of the

same size. And as you took this thing

and you slid it you're sliding it. If

you say sliding it over here one circle

is smaller, one circle is bigger. And

you have some um markings on the

proportional compass that would tell you

say you're getting at uh I don't know

this one's 110, this one's one quarter

and so on. We start going over here,

then we're uh then we're depending on

which which one you're using, you're

enlarging it. So, it was used for

reduplicating a circle, a perfect circle.

this part.

I didn't come in late today, but I was

actually doing some extra preparation. I

thought I was mean meeting to do to help

give you an idea of what things were

like then. Um I have over here this is

somebody put up co-pilot not the

previous one but a new one. Ah yes

it this attitude still exists at first

the I tried to find a good way of

describing it word macho is one way but

macho is gender specific and it also

ties in with someone who's tough

physically strong but there's that sort

of cocky arrogant snoody attitude that

sometimes people have for instance in

the computer lab you might see some if

when you're in first year they say hey

boy you're using a Windows machine real

computer scientists not only don't use a

Windows machine. They're using Linux or

Unix, but they're using command line.

There was a term that they had for

instance um give me a command line or

give me death kind of it has its place.

I'm not anti-command line and actually

you user interface for certain types of

tasks, certain types of users, but for a

typical user like your grandmother, it's

not for everybody. But they say if you

want to do real serless work, you want

to be a real computers, you have to use

a command line. And you, how many of you

have seen this attitude, not just in

computer science, but other ones? They

say a real driver only uses a stick on a

manual transmission. You show hands here

if you heard someone say this sort of

thing. Just to know what I'm talking about.

No. Or maybe Oh, okay. Oh, Jeff. Jeffrey again.

again.

Same person. Oh, few others. Okay. Okay.

A few of you heard it. Just not many.

Look at you. And I was trying to find a

word before class. The macho did it, but

it didn't quite fall out. So I kind of

described it with co-pilot and it

actually did pretty good job. Copy, but

also elitist. You're not a real

computer. I'm a better person. I'm

better driver. Snobby. And here's the

real only use command line interface.

What are some words to describe this? Uh

pretentious. Pretending. See, not that.

Someone who's pretentious. Holier than

thou. Smug. self assured. I never heard

of this. I've never heard this term

here. I suppose this is descriptive. So

why am I bringing this up? This isn't on

the exam, but this will be.

Technology has a lot of uses, but even

say when the desktop calculator came

about, there are some the teachers, they

still might say this, you real people

who who are educated, they don't don't

use the calculator. They're able to do

it themselves. They'll use a tool.

Well, the computational devices that

I've been trying to show you were put to

a lot of real world tasks. They were

very useful.

However, the scholars didn't they viewed

them with suspicion and with cockiness.

Real scholars don't use these devices.

Same sort of attitude when you're

talking about say a graphical user

interface or a automatic transmission.

And some of these, some of them used

them secretly, but they wouldn't admit

to it. And some of them, oddly enough,

we'll see with the what the heck is it?

The slide rule. They invented the slide

rule, but they didn't want to be

associated with it because there is this

negative connotation. That's the

attitude that existed. So the the

computational that's exactly get to

derive it by scholars preferred

basically finding it yourself. William

Alred, there he is the inventor of the

slide rule. This is the way he dis he

characterized these computational

devices. Essentially, you're not a real

teacher. It's all smoke and mirrors.

You're not a real scholar if you're

You get the you understand you get a

feel for the attitudes of people in

those days. Now,

now Renee Dearts. Now, in the notes, I

actually had this in the notes. And he

said it may you need to take down notes

people to understand the context unless

otherwise told it can be on the exam

and I had this it may be clear as I go

through it but by the time you get to

exam it gets hazy that's why I had the

had it frosted out any of the cards

people asked him and he was a known he

was I I read about him I was taught

about him in the magic science and

religion very well-known person a

regarded person but When someone asked

him, "What sort of computational devices

did you use?" What he would show them

was a ruler and a couple of these ones

for measuring angles and one of them was

broken. In other words, he answer

paraphrases, "I don't really use them. I

don't really need them." Someone says,

"Do you use computer?" Someone shows a

Josia method of multiplication. Some of

my students actually have learned this.

Apparently that's what they tell me is

it's still taught in certain in

countries in Eastern Europe where it

originated according to Williams

probable so likely but they haven't had

enough definitive proof to say yes in

these regions so India if you know the

geography India and central Asia these

ones are in western Asia as people

travel and also by Chinese societies so

al uh India

going west to the middle east east

Now with the gelia method, how it works

is this wasn't a device, but it was a

grid to help people perform what for

them complex multiplications. So they

could draw out this grid. Not hard to

just as a reference or reminder. Maybe

I'll put this up here so you have this

available in case you don't. The low

ones you should be able to do in your

head, but the higher ones you might not

have to whip out a calculator. So, I

just quickly

have this handy. So, what I have is

these operands and there's the products.

That's easy to see. These first few you

should be able to do in your head. This

one maybe, maybe not. Now people weren't

as educated even the quote scholars of

the day. So doing something like this

not many people could do.

They might be able to do the simpler

multiplications like this. But you know

with full hand multiplication the 456

times by 128 is basically when you do

the full multiplication I won't do it

here because you should know it from

elementary school. You have the first

row four five six times by one. Then you

push it over four times on six by two.

So you perform these simple uh sorry by

eight. My mistake eight's the smallest

one. 456 by eight. That's for the ones.

And then the four five six by the two.

That's the row below it. But you have a

zero to indicate that's for tens. 456

times by one. And there's two zeros

because that's for the hundreds. I'm

verbally describing what you should know

that technique.

people might be able to perform the

simple multiplications like this

and then do the addition. However, with

the jelloia method of multiplication,

draw the grid and

This is what you draw.

put the B here and be able to draw it

out. How would you do do this with the

simple products? 456 times by one, we

had the 456 here. I'll explain the zero

later, but the zero is for uh carry out.

456 by two,

we have 9 1 2. So they could create it

by the partial products uh four five six

by 8 three a long diagonal at six. So

once you've created this doing the you'd

create this by doing the simple

multiplications and now to be able to

figure out the more complex product you

know that the 456 times by 1 28 it

consists of simpler multiplications that

8 456 by 8 456 x2 46 by 1 and then you

add up those products. So now to be able

to figure out the product we just add

along diagonal. The first well first off

the number that we should have is 58368.

58 368. How do we get that? The first

number 8. Read it off the diagonal. This

one we sum the values on diagonal. 4 + 2

is 6. There's the six. And we continue

on. 6 + 1 uh 7. 7 uh do this. That's 7 +

4 uh 11. 11 +

uh 2 is 13. So, we have a three here,

and it's carried. It's a carry in to the

next column. So, now we add this one

from the previous uh column. 1 + 5 + 1,

that's 7. 7 + 8 is 15. 15 uh + 3 is 18.

So, we have 18 here. But that one,

That's six 10. Uh, sorry. 10.

10.

Let's see what the hell I make a little

bit scary. Oh, sorry.

I added this one completely goofy. 1 + 4

is five. So, there's a five. I got the

old hamster fell asleep for a second.

And zero. So that's how we get the values.

So they could perform the simple

calculations to create the delosia grid.

And once they create it, all they have

to do is read off diagonal to be able to

figure out the more complicated multiplications

multiplications

exam type questions. Let's put some

thought into this. Not as a student, but

let's think about this. If I were making

the if you were making this course and I

was trying to evaluate, do you actually

understand it? I could ask you various

ones that might be kind of tough for the

bark about the explanation of how they

created it. It could be tougher marks as

well. How did they create the how this

work? You described that people used it

um created this part and then they

derived this result from essentially

reading off diagonals. How does it work?

That could be a little bit trickier than

Mark, but it's possible. I could ask you

to do a trace of this device. Um

where using the dillian meth method

evaluate if this device if this grid

created is a legitimate one. Now here's

some thought into it. You probably

wouldn't think about this as a student

but let's step back in the meta as an

instructor. If I gave you this number

and the if I asked you to calculate the

value, it evaluates nothing because if

you have no idea how this glossia grid

this technique works, all I'm evaluating

is can you multiply 456 by 1 28? I'm not

going to ask that question obviously in

postsecary simple multiplication like

this. So what I could do to evaluate if

you understand technique is

the question could be we have a someone

who's drawn out jelloia grid

apply the technique that you were taught

and it determine if this is a correct

grid or not or is this supposed to be

something that's an antiquity and you're

evaluating is it legitimate

according to technique evaluate the

result and I could put basically wrong

results in here. However, I'm looking to

see if you have those wrong results.

Because if you have the wrong results,

if these intermediate multiplications

are off, then it's going to be different

than if you multiply 46.1.2.

But if you get the wrong results and

you're reading it correctly, that tells

me that you understand the technique.

It's highly unlikely that I would ask

you to do multiplication of two numbers

because that doesn't evaluate the

glossia technique. that just potentially

could have evaluate your ability to

multiply. So that's a more tricky type

of exam question. When I ask if I ask

something like this, people during the

exam, they're just asking questions. But

wait, the end product, I'm sorry, it

does it's not right.

And I can't answer that question. I've

already I basically told them what I

told you in class that

I'm evaluating whether they understand

the technique. So I could basically put

a bug in there and purposely see if they

get those wrong numbers, but the wrong

numbers show that they know how this

works. I saw a hand up. I wanted to

finish the explanation here. Emma, and I

see something in chat here. Was that you

in chat? Okay.

Okay. Yeah.

Yeah.

How do you Okay.

How do you pop? Um, should I do it?

Okay, I'll do it once more quickly

because that's complicated. Normally,

I'll say look at the video, but I'll go

through it once more. Essentially, 456

times by 128.

Pull that up. Is the partial products

456 times by 8, 46 * 2, 46 * 1. That's

how you multiply 456 by 1 28. Each one

of these is populated where you write

out the product of 456 times by 1. This

is 456 times by two. 912.

That's how you populate this for the products.

products.

They knew how to populate this by doing

the simple calculations and they didn't

necessarily know how to do the more

complex ones. But once they have this

created, then they can just read off the

And I populate I actually went through

it just to convince myself and I suggest

that you do it. So I do the calculations

here but essentially it's just tracing

another form reading off diagonals. What

do you get the sum and then the

carryover here

when I was first teaching myself the

technique I did this. So I took my old

notes here but as a learner and then I

used them for myself just to make sure

Oh I got to get rid of the annotations. They're

They're

all drawings. Napier's bones. Check the time.

time.

John Napier

know when he's born he he developed

these bones based on glossia method of

multiplication. So for for instance we

got the 456.

It wasn't this one works because it's

just an order snapshot of the order.

If we're say multiplying the 456 by one,

you can see the four five six the first

one row is for the one multiply by one.

Next one is 456 by two and by three and

so on. But we wouldn't keep it like

this. We'd have the jello we'd have the

nap. So napier's bones is based on

glossia method of multiplication and you

can see at the 456 but we had this one

individually created. So, we actually

had each one of these

a strip as a separate column, a separate

each one of these columns is referred to

as a bone like that. And then we'd have

another one

Oh, the zero is kind of a bad example,

but another one that's not necessarily

right beside it, a three. And we have it

for each one of these digits.

And depending on which number that you

were trying to multiply, we rearrange

these. We could have them rearranged

rearranged

to figure out the products. So 30 * 1 is

30. 30 * 2 is 60 and so on. And we could

rearrange it. each one of these separate

bones according to the number that we

were going to perform a multiplication

on. And these partial products would

then tell us we'd sum up the uh sorry in

multiplication we'd sum up the partial

products ourselves but we'd read off the

values of diagonal to figure out the

more complex multiplication. Napier's

bones were

well they're known by different

different names. He wrote a book on it

called the Rabba Deloia. They were also

known by the numbering rods because used

for numbers. They're constructed, yes,

as the name implies, out of bone-like

materials, not bone, but boneike. But

they're also constructed out of other

materials as well, but the bones is the

one that stick. So, it was referred to

as the Napier's bones. Napier for

Napier, the creator of bones for because

originally they were out of a bone-like

material. And they could perform, I've

shown through the glossia method, the

multiplication. It was basically a

physical implementation on bone like

substances um digitalosia method and

they actually took in some later devices

these napier's bones and they would

insert these columns into the

computational device to produce uh the

results of things like a product. So

I've shown you the glossia method so you

can see the multiplication just be aware

they can be used to other computations

logarithms the the square root and the

cube root but you don't have to know

anything other than the multiplications

ten okay usually put something in the

chat so I'll lower your hand see what

you have typed in the chat

no that's the answer you asked how the

cube work and square work you don't have

to know that one I think Williams might

talk about it so if you're interested

you can look at William's book for

instance. This is where I do breath but

not depth. I'm giving you some idea that

these what they can do and I explain one

of them and then if you're really really

interested you can take the time to do

the other ones but I don't want to focus

too much on one just one device because

I want to cover as much as possible.

And truth be told I'm actually a little

behind where I was last in previous

semester. So I want to pick up a bit.

I will show this one a little bit. the

use of Napier's bones just to give you a

little bit of a feel how they work.

recording.

Oh, I think I stopped. I'm going to have

Janelle Lucas rulers. So, this ties in.

the videos here. I add an original one.

For some reason, the audio got

corrupted. And what I have Janelle Lucas

Jenile, the two people came up with it.

He's one and they pronounce it as best I

can in the French way where it's the age

is silent. And then Edward Lucas, you

see this was the background. So, they

were good at numbers. Um, similar with

Napier's bones.

What I'm referring to right here, the

carry part of it is

where you might get screwed up is you

forget with because it's glossia method.

You add along here. This people are able

to figure out these ones, these sums.

But when they get to here, you have that

three carry the one. They're off in

their calculations. They forget to carry

or they carry it when we're wrong.

Sometimes I glitched a bit, but I don't

think I did my calculation off except

for trying to add that as a value.

So, mistakes can be made with

the Janelle UK rulers. Check my time

again. 3 minutes left. It was far

example.

So, say 327. Well, I'll even do the one.

You started right here. Some of these

have multiple values. You read the first

one. Well, this the first one is the

only one. 327 * 1 is 327. You start this

number. That's the first value in the

product. And we follow the arrows. Well,

there's no other choice. This is the

easy one. 327 * 4 is this value.

We go down here and we take the first

value in that row. That first value

points to the next digit, eight. See,

there's an eight. The eight points down

to zero. The zero points to three. The

three points to one.

And that's the value. We just read it

straight off. How we uh so my video

essentially goes through this with

different numbers. I rearrange these

strips and different results. But the

base part is we just come down here to

get um and we start reading the first

value off the first row. and it uses

different examples that covers about

threequarters of the video. The other

part, the extra bonus as it were, um, if

you're interested. Why is it that

there's a lot of videos that actually

describe how to use the Janelle Lucas

rulers and and a lot of resources as

well, not just videos. But what I

couldn't find a few years ago was if

well, if someone actually had had a

resource, why reinvent the wheel? I'd

refer to it. No one actually had an

explanation. I was curious

why is it designed this way? Why is it

actually increased? See 9 to 8 8 1 2. So

it's in sorry decreasing. It's

increasing this way and then it wraps

around the last one quarter of the video

or so basically explains the design of

it. Why is it designed the way it is? So

that you can get the result. That part

won't be on exam, but if you're curious,

start looking through wondering why is

it designed that way? Kind of a clever design.

and we can figure out a division. So

this value this is the um

numerator this is the denominator this

result doesn't evenly divide you have a

remainder of three. So we read off

starting the first volume first row uh 11

1 11 and then down to the five. Five

points over here to the nine and then

the nine points to the three which is a

remainder and we look at the six because

it's this value this combination divided

by six. Oh, one other thing I just in

case you didn't get it implicitly.

Anyhow, this is the multip this is the

division the multiplication. It might

seem useless where okay, we have this,

but what if we want to calculate

something larger? Well, then you could

calculate using the successive

multiplications. Um, say this was 3271

multiply by 41. We calculate the four

part of it. Then we could calculate the

one part of it and then add them

together. So it wasn't just for cases

where one operand was a single digit.

And I think that takes us to time.

Yeah. So there's the videos. We got up

to here.

Let's see how many we have.

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CPSC 409 Sept. 12