Book of Affirmations

Book of Affirmations

Within Book of Knowledge

Here I'm thinking of oaths, vows and credos. At this point, a mere collection.


School Prayer by Diane Ackerman

In the name of the daybreak
and the eyelids of morning
and the wayfaring moon
and the night when it departs,

I swear I will not dishonor
my soul with hatred,
but offer myself humbly
as a guardian of nature,
as a healer of misery,
as a messenger of wonder,
as an architect of peace.

In the name of the sun and its mirrors
and the day that embraces it
and the cloud veils drawn over it
and the uttermost night
and the male and the female
and the plants bursting with seed
and the crowning seasons
of the firefly and the apple,

I will honor all life
—wherever and in whatever form
it may dwell—on Earth my home,
and in the mansions of the stars.

The Healer's Oath

I swear to fulfill
To the best of my ability
To the best of my judgment
This covenant

My highest concern

Will be the well being of my patient

Will be the well being of my community

Will be the well being of all

I will put well being before mere survival

I will respect the knowledge
Passed to me by those who have gone before
And will pass what knowledge is mine
To those who follow

I will inform my patients

Of my diagnoses and reasons
Of preferred and alternative treatments
Of known risks
Of likely outcomes
Of what I do not know

I will obtain my patients consent
I will preserve my patients privacy
I will respect my patients choice of treatment
I will respect my patients refusal of treatment

I will attend my patients
Regardless of their station
With kindness
With consideration
With sympathy
With respect
With integrity
With propriety

I will attend my patients
With my best knowledge
With my best judgment
With my best skills
With awareness of the limits thereof

I will endeavor to do no harm
I will do no less than is needful
I will do no more than is needful

I will ease suffering where I can
I will ease those who wish it from life

I will not withhold treatment
From the poor
From the outcast
From the prisoner
From the stranger
From the enemy
From any



Five Basic RIghts 
(Adapted, blending Universal Declaration of Human Rights with Declaration of Independence)

the right to

1. life 
2. liberty
3. bodily integrity
4. free expression
5. pursuit of happiness
   
   
6. equity
7. equal protection of law
   
8. equal enforcement of law

Rules of Conduct

The Rules of Conduct are based on ethical principles of...

1. honesty
2. integrity 
3. competence
4. service
5. respect
6. responsibility

(King's) Six Principles of NonViolence

1. Nonviolence is a way of life for courageous people.
2. Nonviolence seeks to win friendship and understanding.
3. Nonviolence seeks to defeat injustice, not people.
4. Nonviolence holds that suffering can educate and transform.
5. Nonviolence chooses love instead of hate.
6. Nonviolence believes that the universe is on the side of justice.
   

Principles of Consent (General)

• 
  Voluntary:

  Consent must be given freely, without any undue influence or coercion. 
• Informed:

  Individuals must be adequately informed about what they are consenting to, including potential risks and benefits. 
• Capacity:

  The individual must have the mental capacity to understand and communicate their consent. 
• Specific and Current:

  Consent must be specific to the activity or treatment in question and given at the time it is needed. 
• Ongoing:

  Consent must be actively maintained throughout the activity or treatment and can be withdrawn at any time. 

Principles of Consent (Medical)

• Capacity:

  The individual must have the mental competence to understand the information presented and make a reasoned decision. This includes having the ability to understand the risks and benefits of the situation, and to appreciate the consequences of their decision. 
• Disclosure:

  The provider (e.g., doctor, researcher) must provide clear and comprehensive information about the situation, including the nature of the procedure or intervention, the risks and benefits, and any reasonable alternatives. 
• Comprehension:

  The individual must be able to understand the information provided. This means they can grasp the meaning of the information and how it applies to their specific situation. 
• Voluntariness:

  The individual's consent must be freely given, without coercion or undue influence. They must have the freedom to choose whether or not to participate or undergo the procedure. 

Book of Number

 Book of Number

Forewarning

beware what you count
you may come to value only what counts

number is clean
life is not

what fits in digits
leaves much behind
what’s left behind
can be what matters

not all things can be named in number
not all truths can be measured
not all measures tell the truth
not all truths can be proven

yet with number
we carve sky from sea
predict the falling fruit
and follow stars home

so reckon well
but reckon with care
number is sharp
and sharp things cut

First Things

we see one
we see another
we see the same

we match what we see
to what we know
and give it a name

we count
to remember
to compare
to tell

but not all things are counted
some must be measured

we count fingers
we measure time

we count stones
we measure distance

we count flocks
we measure water
and cold
and heat

number is not what we have
it is what we know
what we tell
what we show

counting makes steps
measuring makes lines

and from these lines
we make a tool

we call it the number line

it begins with one
then two
then three
and up we go
into the many

but there is a space
between what is
and what is not

a space with a name

zero is not nothing
but the name of nothing

a mark
a breath
a pause in the beat

a place where something could be
but isn’t

zero is the fulcrum
between more and less
between gain and debt
between warm and cold
the point where water freezes

below zero
we find the cold
the absence
the pull

we name these
negatives

not because they are bad
but because they turn the other way
measuring from presence
counting from absence

number runs both directions
outward
from zero
into the more
and into the less

number relates to number
each to all others
counting the distance between

Number

we count with symbol
we count with name
a finger to each

0 zero no finger
1 one a finger raised
2 two a pair held wide
3 three a grasp with space
4 four the claw of paw
5 five the open hand

6 six five and one
7 seven five and two
8 eight five and three
9 nine five and four

a finger remains
its number ten
but may we not open symbol
to count toward infinity?

Place

10
ten
the last finger

1 and 0
ten and none
side by side
they name the next

where each number sits
tells power

leftward
each place is ten times more
rightward
each place is ten times less

between tens and tenths
we set a dot of meaning
marking the place of ones

the fulcrum of number
between the great
and the small

00.0
is zero tens
and zero ones
and zero tenths

11.1
is one ten
and one one
and one tenth

99.9
is nine tens
and nine ones
and nine tenths

what then is 999.99?

how does a shift of the dot
left or right
transform a number?

with only ten numbers
marks from zero to nine
and one small sign
we count the infinite

from infinitely large
to infinitely small
in infinitesimal steps

place
is the magic
of infinity
counted from fingers

Fractions Ratios Degrees

draw a circle
whole and round
without beginning
without end

slice it
whole into halves
halves into quarters
quarters into eighths

1 → 2⁄2
1⁄2 → 2⁄4
1⁄4 → 2⁄8

1⁄2 one of two
2⁄4 two of four
3⁄8 three of eight
we name these fractions

fractions are written a⁄b
a parts of a whole
cut into b pieces

fractions
divide a whole
to name the pieces
to count the pieces
in relation to that whole

a ratio
is a comparison
between two things
like fractions
yet unlike

ratios are written a/b
a things to b things
a things per b things
a things divided by b things

this to that 
5 fingers per hand
1 hand to 5 fingers
10 fingers to two hands
10 fingers / 2 hands

how does order change use?
how does meaning change use?

a⁄b
can mean a part of a divided whole
or a relation between two things

a fraction of the circle
a ratio of the pack

some ratios stay simple
1⁄2 = 0.5
3⁄4 = 0.75
3⁄8 = 0.375

some stretch on without end
1⁄3 = 0.333…
and endless thirds
the closer we look

***

circle back

draw a circle
and slice it into 360 pieces

we name each piece a degree

we mark degrees  
with the ° symbol  
to count slices of turn  
fractions of a full spin
pointing direction
measuring angle

90 degrees = 90° = 90°⁄360° = 1⁄4 of a circle
3/4 of a circle =  3 x 90°/360° = 270°⁄360° = 270 degrees

***

circle back

draw a circle
mark a point along it
measure its run
point back to point
we name this circumference

measure straight
from point to center
we name this the radius
continue to its opposite edge
twice the radius
to find the diameter

what is the ratio of round to span?
of circumference to diameter?
what is this mystery?

not a whole
not a fraction
not a number that ends
not a number that repeats

we name this ratio pi
we give it a symbol π

 if you would know number follow pi

Power and Root

to raise to a power
is to stack
the same number
upon itself

2 × 2 = 2² = 4
2 × 2 × 2 = 2³ = 8
10 × 10 = 10² = 100
10 × 10 × 10 = 10³ = 1000

powers build
slow at first
a quickening
then leaping away

***

a point
has no dimension
any number raised to 0 power
is one alone
n⁰ = 1

a line
is a point given length
any number raised to 1 power
is itself
3¹ = 3
a row of 3

a square
is a line given breadth
3 × 3 = 3² = 9
a grid of 3 rows and 3 columns

a cube
is two dimensions
square plus depth
3 × 3 × 3 = 3³ = 27
a block of 3 by 3 by 3

a tesseract
is 3 dimensions plus one
cube plus a shift
3 x 3 x 3 x 3 = 3⁴ = 81
a hyper cube of 3 by 3 by 3 by 3
a cube over time?

but not beyond number

what may we not
build as a shape?
what may we not
build as a matrix?

***

number flows away from zero
less as well as more
what of negative number
when applied as powers?

10^(-1) = 1/10¹ = 1/10
10^(-2) = 1/10² = 1/100

falling away
to infinitely small

***

to seek a root
is to ask
what number
built this power?

the root
is the side of the square
is the side of the cube
is the side of the tessaract
and beyond

we set root degree a
before root symbol √
and number last b
a√b

the square root of 25
25^(1/2) = 2√25 = 5
because 5 × 5 = 5² = 25

the cube root of 27
27^(1/3) = 3√27 = 3
because 3 × 3 × 3 = 3³ = 27

roots unbuild

***

powers build
roots unbuild

yet they are bound
one to the other

some as whole number
some as fractions
some as ratio

the square root of 10
10^(1/2) = 2√10
the cube root of 8
8^(1/3) = 3√8

a ratio of powers
is a root
is a fraction of powers
is a power

49^(1/2) = 2√49 = 7

power and root
root and power

***

some powers stretch
toward infinity
toward infinitly small

we count
and measure
and compare

farther than we can count
farther than we can grasp

what is the difference
between countable infinity
and that infinity
infinitely divided?

***

draw a triangle
with legs at right angle
call them a and b
opposite a long diagonal
call it c

if a = 3
if b = 4
if c = 5

does a² + b² = c²?
is this true for any value?
how would you know?

***

draw a square
of side one
sides a = b = 1

draw a diagonal
corner to corner
name it c

what is the length
of that diagonal?

c² = a² + b²
c² = 1² + 1²
c² = 2 
c = 2√2

try as you may
you will not find
any ratio of number
equal to 2√2

a⁄b ≠ 2√2

this number runs on
without repeating
without end

irrational
and yet real
between the corners
of a simple square

what other numbers
what other kinds
will you discover?

Operators and Order

when we reckon number
speak with care
speak in order

some reckon first
some wait their turn

we agree on order
to reckon true

***

some symbols hold others
do these together
do these first

( and )
we name parentheses
cupped hands
holding a sequence
to be taken first
to be taken as one

***

some symbols are instructions
do this
do that

we call these operators

some are paired
each a reflection
one of the other
complementary
dancing a unity

 + and −
plus and minus
add and subtract

× and /
times and per
multiply and divide

^ and √
raise and root
exponentiate and extract ... find the root?

***

not all steps
are taken in the order written

this is the order of operations
the steps of the dance

p parentheses ( )
e exponents ^
m multiply ×
d divide /
a add +
s subtract −

pemdas
an order of intention

***

look

2 + 3 × (4 + 1)² / 5

goes not
left to right
but deep to shallow

first
(4 + 1) = 5
then
5² = 25
then
3 × 25 = 75
then
75 / 5 = 15
finally
2 + 15 = 17

***

without rules
a scribble of symbols
means many things
without agreement
meaning is lost

not law but language
not nature but pact

we reckon
in common order
to find common truth

Logarithms

winds rise
but not in straight line
we name them
on a scale
that rises exponentially
in magnitudes of power

a gentle breeze
rustles the leaves
a strong breeze
shakes the branches
a gale
bends the trees
a storm blows them over

each increase
not in equal measure
but in power raised

feel the wind
count its strength
not in measure
but in magnitude

***

some sequences
grow number
step by step

add one
add one more

1, 2, 3, 4, 5, 6...

counting arithmetically
in even steps
each added
to the one before

regular
and even

***

some sequences
grow number
leap by leap

raise one
raise one more

10⁰ = 1
10¹ = 10
10² = 100
10³ = 1000
10⁴ = 10000...

counting geometrically
in growing bounds
each a multiple
of the one before

we name such leaps
orders of magnitude
each leap
a power of ten

***

to ask
how many times
must we raise ten
to reach a number
is to ask for its logarithm

we call it log for short

log₁₀(1) = 0
log₁₀(10) = 1
log₁₀(100) = 2
log₁₀(1000) = 3
log₁₀(10000) = 4...

to ask
how many times
must we raise two
to reach a number
is to ask for its logarithm

log₂(1) = 0
log₂(2) = 1
log₂(4) = 2
log₂(8) = 3
log₂(16) = 4...

logarithms are leaps
expressed as steps

the shadow of exponential growth
cast onto a number line

***

on a ruler
each mark
lies equal distance
from its neighbors

1, 2, 3, 4…
equal spacing

take two rulers
number lines
laid side by side
offset one from the other

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10...
   0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10...

two and zero align
five and three align
with distance we add
with distance we subtract
one scale measured
against its twin

***

but a logarithmic scale
does not step numbers evenly
but in leaps

between 1 and 2
a wide step
between 2 and 3
a shorter one
from 9 to 10
a mere breath

equal distance
here means
equal ratio
between positions
spaced in leaps

1, 2, 3...
are no longer evenly spaced

1 to 2
2 to 4
4 to 8
equal steps
of multiplication

this is the ruler
of logs

***

logarithms turn
multiplication into addition
division into subtraction

they let us measure ratios
as distance

leaps
counted in steps

***

sliding rulers compare
not two points
but four
a aligned with b
c aligned with d
each pair offset
by scaled distance

three are known
and set to match
the fourth is revealed
in answer to the question posed
reckoning value
from distance along a scale

two numbers here
two numbers there
the distance between
one pair
matches the next

on a straight line
distance is difference
(b − a) = (d − c)
adding and subtracting

on a log line
distance is ratio
(b / a) = (d / c)
multiplying and dividing

how we slide
how we group
tells us
what we find

***

logarithms are the inverse of powers
where powers leap
logs count steps

ask now

what are the fractional values
of logarithms?

what is log₁₀(5)?

what might this number describe?

***

what happens
when we add logs?

what is
log₁₀(1) + log₁₀(10)?

what happens
when we subtract logs?

what is
log₁₀(100) − log₁₀(10) = ?

what might these numbers describe?

***

what happens
when we divide logs?

what is
log₁₀(10) / 2?
what number
has a log of 0.5?

what happens
when we multiply logs?

what is
2 × log₁₀(10)?
what power
does it become?

what might these numbers describe?

***

what is the spacing
along a logarithmic ruler?

why does spacing shrink
as values grow?

if you slide two such
side to side
what means distance?

what might be accomplished
sliding one scale
against its twin?

***

if logs are steps
and powers are leaps
how might one measure the other?

what might be raised
what might be rooted
from rulers of ratio?

***

logarithms
untangle powers
flatten growth
reveal the steps
behind the leap

to see clearly
when things grow fast
when things span far
when things change
by orders not steps

we turn to logs
to measure what swells
to track what accelerates
to name
the thunder
in the storm

Form and Function

what goes in
determines
what comes out

one to one
we name a function
set to set
we name a functional relationship

***

a function relates
members of one set
to members of another
in one way only

we call this mapping

each set of inputs
to a set of outputs
and no other

like parents to number of offspring
like distance traveled to time and speed
like hours of waking to hours of sleep

***

within each function
is a rule
a method
a process
an ordering of instructions

we call this
an algorithm

the algorithm is the box
the function is the bridge

some algorithms are short
a single step
some algorithms run long
ranging far and wide

if the steps are clear
and always end
the rule can be followed

what if  an algorithm
can end but does not?
what if an algorithm
cannot end?
how can we tell
if it will end?

***

a function is not
the numbers it acts on
nor the steps inside
it is the
relationship itself

how one thing
determines another

anything
that takes input
and gives output
by a rule

***

to see a function
we plot its path
in an ordered field

one line horizontal
we name the x axis
one line vertical
we name the y axis

crossing at
the zero of each
positives count upward and right
negatives count downward and left
a pact of convention

this flat field
we call a plane
we name this plane
a graph

***

to plot a point
we take two counts
one along each axis
x and y
(x,y)

so far over or back
so far high or low
a relation of two

one by one
point by point
we draw the shape
of relation

a point
a curve
a wave
a climb
a drop
a broken step
a sudden stop

each function
its shape

***

a third axis
brings depth
new possibility
relations of three

a volume forms
where once was plane
a space for every
x and y and z

what shapes are possible
in this deeper space?

***

what if we add another
axis upon axis?
how far might we go?

how many paths
can cross and twist
in space
beyond our seeing?

what if each point
holds many answers?

***

some scales count
in even steps
others grow
by leaps and bounds

what happens
if we scale an axis
logarithmically?

what shapes might we reveal?
what might they mean?

Algebra

we begin with constants
what is known
what is guessed
what is chosen
what is steady in the wind

Number is constant
0, 1, 1.5, 1/3, 3√8

some numbers
are given symbol

π ~ a circles circumference to diameter
c ~ the speed of light
g ~ the pull of Earth

all are constant
their value fixed

***

when number is unknown
we devise a vessel
we name a variable

we use a letter

x, y, z
a, b, c
A, D, C

to hold what is
to hold what may be discovered
to hold what may be assigned

standing in place of what is unknown
standing in place of what might vary

***

we build expressions
with constants
with variables

x + 3
2a
y × y
d / t
πD

expressions are forms
not yet rules
not yet questions

***

we balance expressions
setting one
= equal to the other
we name these equations

1 = 1
a = 2
π = C / D
2ab = 2 × a × b
(a + b)^2 = a^2 + 2ab + b^2

one side balanced
against the other

some equations are true
1 = 1
some equations are false
1 = 2

some equations are true
but only for some value
x/1 = 1
where x = 1

some equations are true
only for certain values
x^2 = 1
where x = (1, -1)

an equation
is an assertion
of balance that one expression equals another in common value

If one side
balances the other
we call this true

if one side does not
balance the other
we call this false

some equations are undefined
x / 0 = undefined

why is x / 0 not infinite? x/x =1 for all numbers with one exception which is excepted?

***

some equations
assert a truth
no matter the values

addition is commutative
a + b = b + a
and associative
(a + b) + c = a + (b + c)

multiplication is the same
a × b = b × a
(a × b) × c = a × (b × c)

but subtraction is not
a − b ≠ b − a
(a − b) − c ≠ a − (b − c)

nor division
a / b ≠ b ÷ a
(a / b) ÷ c ≠ a ÷ (b ÷ c)

when order matters
keep the order

yet

subtraction is
adding a negative
a − b = a + (−b)

division is
multiplying by a fraction
a / b = a × (1 / b)

a root is number
raised to a fractional power
2√a = a^(1/2)

every doing
has its undoing

every action
has a reversal

when subtraction is addition
is it commutative?
is it associative?

when division is multiplication
is it commutative?
is it associative?

***

some equations assert a balance that holds only when variables have certain values we name these conditional to solve an equation
is to find the unknown
is to make the variable
stand alone balanced with value

x + 3 = 7
subtract 3 from both sides
x + (3 − 3) = 7 − 3
x = 4

we subtract 3
to isolate x
we subtract
from both sides
to preserve the balance

when x is buried deep
this may require many steps
on the path to solution

***

2x = 10
divide both sides by 2
2x / 2 = 10 / 2
x = 5

each step
a doing which undoes
each step
preserves balance

this is algebra's power

***

what is x?
what does it count?
what kind of thing
do we mean?

10 of what?

number alone
is a ghost
a shape
with no substance

a unit is one kind of thing

we think in units
we label with units

some are a count
14 days
5 fingers

some are ratios
asserting a relation
true when inverted
28 days / 1 lunar.month
1 lunar.month / 28 days

these expressions
form equations
5 fingers / 1 hand x 2 hands = 10 fingers
(1000 paces / 1 day) x 10 days = 10000 paces

units combine when we multiply
2 persons x 5.hours = 10 person.hours
with a ratio of
2.5 hours / 1 person

how might we use
such a notion?
do 10 hour.persons make sense?

units only mix with meaning
yet some meanings surprise
odd pairings expand the mind

units reduce when we divide
10 person.hours / 4 persons = 2.5 hours

numbers are clear
yet the world is complex

If a task requires 10 person.hours
how long will 1 person require?
how long will 10 persons require?
how long will 100 persons require?

are all numbers practical?
can other numbers
other ratios
help us predict?

***

these are bare bones
in the bone setters art

to find the unknown
from hints
from method
from relations in flux

is a mystery at the heart of number

if you would find number
follow algebra

Algorithms

not all questions
are solved at once
some unfold over time

not all answers
are stated in balance
nor found in balance
some emerge by doing

an algorithm
is a method
is a program of action
is a set of steps to follow

a function
that runs a path
from inputs to outputs
among the myriad possibilities

each step followed
each outcome built

some algorithms
are simple
a straight path
from input to result

some reduce
a complex task
into simpler tasks
bites which may be chewed
each small answer
a piece of the whole

we set an algorithm a task

it answers
if the steps are clear
if the steps are correct
if the task may be completed

are there tasks
which cannot complete?
how do we know them?
can we know beforehand?

***

a variable
in algebra
is a mystery

a variable
in algorithms
is a container

we assign it
we change it
we use it again

algebra asks
what is x?

algorithms command
what x shall be
how x shall change
when x shall change

***

order matters

some steps must come
before others
as one lays a fire
small to large

what must happen first?
what depends on what?

what must be done
what must be known
before decision?

***

some paths repeat

while this is true
keep doing these

until this is true
keep doing these

we name these loops
we name looping iteration

some paths branch

if this is true
then do these things
else do those things

we name these tests of truth
conditionals
if then else
while
until

***

consider a sequence
where each step
builds the next

start with 1
and 1 again
each number after
is the sum of the two before

1, 1, 2, 3, 5, 8, 13...

what sequences result
when starting from
some other two numbers?
the same or different?

where do you see these sequences
occurring in the world around?

this algorithm is iterative
repeating a loop of doing

1. let a and b be two values
   
2. write a and b
3. let c = a + b
   
4. let a = b
   
5. let b = c
6. write c
7. until weary repeat from step 3
   
8. end

this algorithm is recursive
calling on itself for answers

let the rule
to find the number
at position n

be as follows

if n is 1

return the first value

if n is 2

return the second value

else

call the rule again

once with n − 1

once with n − 2

add the answers

and return that sum

each recursion
calls on itself twice

asking a simpler question
until what is known
returns

answers
rise back
like bubbles
from the deep

iteration repeats the steps
recursion repeats the pattern

iteration walks the loop
recursion calls itself 
asking less of itself
with each call

***

an algorithm is not magic
but memory and motion

can a set of rules build a path?
can a set of rules find the shortest way?
what if the river rises?
what if the steps take too long?

what makes an algorithm effective?
what makes an algorithm efficient?
what makes an algorithm good?

if you would
shape questions into method
shape method into action
shape action into answers
follow the algorithm

---

Geometry

---

What Is Geometry?

to measure the world
we must see it first

see space
see form
see distance
see shape

we draw a line
we mark a point
we sketch a curve
we enclose a shape

we name these shapes
to name what we see

we study their properties
to learn what they mean
to learn how the world is shaped

we call this geometry
geo-metry
earth-measure

---

Dimensions

all things we know
have three dimensions in space:
length
breadth
height

we may imagine them without
that we may learn from pure form
we reason them with more
that we may learn possibility

imagine a point—
a mark with no size
a mark with no dimension
a location
a corner
an intersection
a world unto itself

imagine a line—
one dimensional
a stretch in one direction
a world of two directions

imagine a square—
two dimensional
length and breadth
at right angles
an area within a flat
we call a plane

imagine a cube—
three dimensions
length and breadth and height
at right angles
each to all
a volume within an expanse
we call space

reason a hypercube—
four dimensions
length and breadth and height and one
at right angles
each to all
a space we cannot see
but may yet conceive
a hypervolume within a construct
a we call hyperspace

how many corners?
how many edges?
how many faces?
what is its volume?
what is its use?

what cannot be sensed
may nevertheless
be imagined with reason

how many dimensions
might we imagine?

---

Postulates and Proof

some truths
we propose as given
we accept them without proof
we build upon them

we call these postulates

the shortest path
between two points
is a straight line

all right angles
are equal

a circle
can be drawn
from any center
with any radius

what can be made
after these truths?
what must be true
if these are true?

we call this a theorem
and build its truth
from steps
from reason

we call this a proof

what follows must follow
if our postulates are sound
if our reasoning is sound
 

---

Point at the Center

mark a dot
you mark a point

a point has no size
no length
no breadth
no depth
no dimension
only position

extend that point
you make a line

slide that line
you make a plane

raise that plane
you make space

one point defines a point
two points not the same define a line
three points not in line define a plane
four points not in plane define a space

this imagined world
our experienced world
one a model of the other

begins with a point

---

Line and Straightedge

a straightedge
draws what thought defines

fold a space
with edge and compass
and something new appears

mark a point
draw a line through the point

how many lines
can be drawn through that point?
how many planes?

mark a second point
draw a line through both

how many lines
can be drawn through both points?
how many planes?

how many points
define a plane?

line unbounded
we call line

line extended from one point
we call a ray

Line bounded by two points
we call a line segment

two lines meet
at one point
in one plane

two planes meet
along one line

we say they intersect 

---

Angle

mark a point
draw two rays from it
the V between them
we call angle

mark a point
draw a line through it
forming two rays

from one ray to the other
we divide into 180 equal angles
we call each angle one degree
we mark each degree 1°

from one ray back to itself
we call 360°

we call 90°
a right angle

we call less than 90°
an acute angle

we call more than 90°
an obtuse angle

when two lines intersect
they define one plane
they form four angles

how do adjacent angles relate?
how do opposite angles relate?

---

Triangles

three points
defining a plane
joined by three line segments
we call a triangle

in every triangle
the sum of angles
is always 180°

how might this be proven?

a triangle with one angle of 90°
we call a right triangle

a triangle with equal sides and angles
we call an equilateral triangle

a triangle with two equal sides
we call an  isosceles triangle

a triangle with no matching sides
we call a scalene triangle

how might we use each kind?

---

Rectangles and Squares

four points
in a single plane
joined by line segments
each at right angles to the other
we call a rectangle

a rectangle with sides
of equal length
we call a square

join to opposite points
with a line segment
two right triangles are formed

how do triangles and rectangles relate?

what may we prove
one with the other?

---

Circles

mark a point
from it
draw a line segment of length R
swing this segment around the point
back to where it began

the shape you have drawn
we call a circle
drawn with a compass

C
the length of the circles edge
from point around to itself
we call circumference

R
from circles center to edge
we call radius

D = 2R
through circles center
from edge to edge
we call diameter

ℼ = C / D
the ratio of circumference to diameter
we call pi

A = πR^2
the amount of plane
occupied by the circle
we call area

---

Polygon and Polyhedron

a planar shape of many sides
enclosing area
we call a polygon

what polygons may be
beyond triangles
beyond rectangles
as we add sides?

we may name each
for the count of their sides

for a polygon of n sides
their sum of interior angles
is equal to (n -2) x 180°

if their angles are equal
we call them regular

a spatial shape of many faces
enclosing volume
we call a polyhedron

what polyhedrons may be
beyond tetragons
beyond cubes
as we add faces?

we may name each
for the count of their faces

if faces and angles match
we call them regular

is a regular polygon
with infinite sides a circle?

what shapes may be folded
to enclose a space?
what shapes cannot?

how might we go on
in further dimensions?

---

Point and Straightedge and Compass

mark a point
construct a circle
extend a line

with point and compass
compass and straightedge
you construct the world
with rule and reason

how might we copy a length?
how might we divide it?
how might we bisect an angle?
how might we construct a shape?
how might we relate shapes?

what might we prove?
what cannot be proven?

what is the difference
between what is drawn
and what is proven?

---

Congruence and Similarity

some forms are the same
in size and shape
we call them congruent

some forms are the same
in shape but not size
we call them similar

a small triangle
a large triangle
if angles match
and sides scale evenly
then they are similar

---

Transformation

shapes may be moved
without being changed

we call this transformation

slide a shape
without turning it
we call this translation

turn a shape around a point
we call this rotation

flip a shape around a line
we call this reflection

resize a shape in constant proportion
we call this scaling

if shape remains
and angles stay true
we say the form is preserved

---

Area and Volume

area is how much flat
volume is how much fill

a line has length
why?

a rectangle has area
length × width
why?

a triangle has area
base × height / 2
why?

a circle has area
π × r²
why?

solids have volume
a cube has side³
a cylinder has π × r² × height
a sphere has (4⁄3) × π × r³
why?

as we add dimensions
how do these relations progress?

---

Coordinates and Proof by Drawing

place a grid
over the plane

mark each point
with an (x, y)

this is coordinate geometry
number meets shape

y = mx + b
a line with slope m

prove a shape is square
by sides and angles

check distances
check angles
show what is known

reason with drawing
and algebra
and postulates

to know what is true

---

What Else May Be?

 can parallel lines
ever meet again?

what if they diverge?
what if the converge?

what does this mean
for the shape of space?

what of shapes
that shift with time?
can time be a dimension?

what is a shape
in four dimensions?

how long is a shoreline?
does it matter how fine we measure?
what is its dimension?

what may be imagined?
what may be reasoned?
what may be drawn?
what may be proven?

what may not?

if you would know form
if you would know structure
follow geometry

---

Trigonometry

a triangle speaks truth
to those who listen

with one simple shape
we can find
what we do not know
from what we do

in physics and sailing
in architecture and astronomy
in shadows and signals
the relations among these parts
show us the world’s shape

What Is an Angle?

draw a point
draw a circle around it
draw a radius
draw another

how might we measure it?

the curve between them
along the circle’s edge
we call an arc

how long is the arc?
how might we measure its angle?

---

Degrees and Radians

we may divide a circle
into 360 equal parts
we call them degrees

why 360?
how many ways
can 360 divide?

we might measure by degrees
in whole and in part

or
measure a length of one radius
along the arc of the circle
the angle it sweeps
we call one radian

a full circle’s arc
measures 2π radians

as circumference C = 2πR
and in a unit circle R = 1
so C = 2π × 1 = 2π radians

how many degrees in one radian?
how many radians in a right angle?

to hold an angle’s measure
in degrees or in radians
we use the symbol θ
we say the symbol theta

degrees divide evenly
radians fit naturally
each has its moment

when might we prefer degrees?
when might we prefer radians?

---

The Unit Circle and Right Triangle

place a grid
across the plane
draw the x-axis
draw the y-axis
mark the origin
at (0, 0)

draw a circle
with radius one
centered at the origin
we call this the unit circle

to construct a right triangle
in standard position
on the unit circle

draw a radius
a line from the origin
to a point on the circle

its length is unitary one
we name this side the hypotenuse side

from that point
drop a line straight to the x-axis
we name this the opposite side

draw the horizontal line
from that foot to the origin
along the x-axis
we name this the adjacent side

we now have a triangle
with one right angle
and one angle at the origin
we name this angle θ
we name this angle theta

hypotenuse
opposite
adjacent
theta

their relations
and their traces on the circle
are the heart of trigonometry

---

Trigonometric Functions

consider our right triangle
on the unit circle

trigonometric functions
relate θ to ratios of its sides

we build these functions
from the ratios of sides
found within the unit circle

sine is opposite / hypotenuse
we write it as a function of θ
sin(θ) = y / 1 = y

cosine is adjacent / hypotenuse
we write it as a function of θ
cos(θ) = x / 1 = x

tangent  is opposite / adjacent
we write it as a function of θ
tan(θ) = y / x

cotangent is adjacent / opposite
we write it as a function of θ
cot(θ) = x / y

secant is hypotenuse / adjacent
we write it as a function of θ
sec(θ) = 1 / x

cosecant is hypotenuse / opposite
we write it as a function of θ
csc(θ) = 1 / y

these are the tools
to find unknowns
wherever right triangles
are found
scaling upward
from unitary ratio

---

may not every triangle
be seen as two right triangles
joined?

how do functions relate
one to another?

how do functions plot
as θ changes value
sweeping the circle?

what repeats?
what reflects?
what returns?

what if θ runs beyond 2π?
or before zero?

do any values repeat?
how often?
what changes?
what stays the same?

what if we leave the plane
exchange the circle for a sphere?

look outward
where are triangles found?
where does angle matter?

look to things at known distance
how might we find their height?
their separation?

look to things of known height
how might we find our distance off?

look up from our world to the stars
how might we find our place?
how might we find our way?

if you would know
the triangulation of structure
follow trigonometry

---

Calculus

some change is steady
like the moons traverse
across the sky

some change shifts
like gusts of wind
sharp and sudden

how might we measure movement?
how might we measure change?

value at an instant
tells you what is

rate of change
tells you what will be.

calculus is the measure
of motion and change
reduced to still fractions


A Fish for Dinner

a fish lies upon a cutting board
its body curves from head to tail
changing height
in measured steps
along its length

we slice this curve at intervals so wide
each height varies with the curve of flesh
each piece has area
width times height

If we slice thick
the cut is hefty
the curve is crude

if we slice thin
the cut is spare
the curve is fine

if we slice infinitely thin
the cut thins toward nothing
the curve is itself

what may we deduce
from rise over run
at each slice?

we clap our hands
we sum the bits together
we rejoin its slices
and the fish is whole again

what is more
the slicing and the summing
are the same process
run to and fro

would you know this magic?

---

Functions

extend your hand in sunlight
with fingers spread
observe its shadow

how many fingers
does your shadow hold forth?

sunshine is
the rule
the function

your hand and fingers
spread as you chose
offering to the rule
the input

your shadow
the result
the output

wriggle your fingers
wave your hand
how does its shadow vary?
what is the rule of light?

we write f(x) = y
we say f of x equals y
we mean rule f applied to x results in y
we mean rule f given input x outputs y
we mean given rule f then y varies as x
we mean given rule f then f(x) and y are interchangeable

f(x)
x and y
each are values
we call x independent
we call y dependent



Limits

a wall is a limit

we call a value approached
in infinitesimal steps
the limit

a limit is a number approached

we write limᵣ→0
we say the limit as r approaches zero

we write limᵣ→∞
we say the limit as r approaches infinity

we write limₐ→b
we say the limit as a approaches b

when number is a function
we write limₓ→r f(x)
we say the limit of f(x) as x approaches r

we ask in this way
as x approaches r
what does f(x) approach?

What happens as
limₓ→0 f(x)
where f(x) = 1/x?
does f(x) ever equal zero?
is f(x) infinitely large?
is f(x) infinitely small?
is f(x) undefined?
is f(x) all of these?

as x nears zero
f(x) diverges without bound
the answer lies in approach
not arrival

much may be seen
from the border of vanishing

limits are the doorway
to the infinitely small
to the infinitely large
to the infinitely many
to the infinitely few

they reach the world
by never arriving


Infinitesimals

approach a wall
halfway
then half again
then half again forever
we approach ever nearer
yet never arrive

how close can we get
to some value
without arriving?

calculus is a language
of becoming
of tending toward
of approach

we approach
in steps shrunk infinitely small
yet never zero

we arrive
at distance which is no distance
a quantity which is no quantity

this distance which is no distance
this quantity which is no quantity
we call an infinitesimal

we use the symbol d
we say the symbol dee
to indicate unmeasurably small change
to indicate infinitely small change

this infinitesimal step
is the giant leap
to motion and change

Differentials

an infinitesimal difference
an interval which is an instant
an interval which is not zero
in a variables value
we call its differential

for differentials
we write dx
we say dee x
we mean the differential of x
we mean an infinitesimal change in x

we reach the differential
as the limit of approach

we reach the differential dx
as Δx tends toward zero

dx is neither zero nor x

we write f(x) dx
we say f of x dee x
we mean f(x) * dx

we mean the product of f(x) and dx
we mean the product of y and dx
we mean a rectangle f(x) by dx
we mean a rectangle y by dx

for any x
we mean this rectangle at x
we mean its area at x
we mean the amount at x

an infinitesimal change in x implies
an infinitesimal change in y

we write dy
we say dee y
we mean a differential of y
we mean an infinitesimal change in y

dx is increase in run
squeezed toward nothing
dy is difference in rise
squeezed toward nothing

yet in dy / dx
proportion remains
slope remains
rate of change at x
is known



The Derivative

some curves rise
some fall
some pause at peaks
or valleys

how do we measure
their rate of change
at any given instant?
for any given value of x?

a line that kisses a curve
touching one point alone
and no other
we call this line tangent

we use the symbol 𝝙
we say the symbol delta
to indicate measurable change
to indicate finite change

the steepness of a line
its rise over run
its change in y
over change in x
its 𝝙y / 𝝙x
we name this its slope

the slope of a curve
is that of its tangent
measured at a single point x
as dx becomes infinitesimal
as 𝝙x becomes dx
as 𝝙y becomes dy
as derived from the curve
as derived from its function f(x)

the limit of Δy/Δx as Δx approaches zero
the limit of Δy/Δx as Δx approaches dx
the limit of Δy/dx as Δy approaches dy

this slope ratio of dy / dx
we call the derivative
the curves rate of change at x
the shape beneath the shape at x

we define the derivative
as the limit of the ratio
as Δx approaches zero

let Δy = (f(x + Δx) – f(x))

we write limΔx→0 Δy / Δx = dy / dx
we say
the limit of delta y over delta x
as delta x approaches zero
equals dy / dx

we call derivation
the process given f(x) of
finding the rate of change at any x
finding dy / dx for any x

we write d/dx
we say dee dee x
we mean the derivative operator
we mean take the derivative of what follows
with respect to x

d/dx is a command
dy/dx is its result
one asks the question
the other answers

we write d/dx f(x)
we say dee dee x of f of x
we say the derivative of f(x) with respect to x
we mean the slope of f(x) at x
we mean the rate of change at x
we mean the derivative of f(x)

we derive a related function f’(x)
we write f’(x) = d/dx f(x)
we say f prime of x equals dee dee x of f of x
we mean f’(x) is the derivative of f(x)
we mean f’(x) is the slope at x
we mean f’(x) is the rate of change of y
with respect to x

we write f’(x) = dy / dx
we say f prime of x equals dee y dee x
we mean f’(x) is the slope at the instant of x
we mean f’(x) is the instantaneous slope of x
we mean f’(x) is the rate of change at x
we mean f’(x) is the derivative of f(x)
we mean f’(x) is the derivative of y

what of y?
as x changes
how does y change?

y = f(x)
we say y equals f of x
we mean y is the result of the rule applied to x
we mean y is the function of x
we mean y is the height at x

dy = f’(x) dx
we say dy equals f prime of x dee x
we mean dy is the infinitesimal change in y
as x changes by an infinitesimal dx
we mean dy is the change in y
applying the rate of change at x
over the interval dx

a derivative f'(x) is a rate of change
a derivative f'(x) is a rule of change

f’(x) is a rate
f’(x) is a slope
dx is an amount of run
f’(x) dx is a rule applied to dx
dy is an amount of rise

f’(x) dx is a product
slope times run
(rise / run) * run = rise
dy is the result

dy is the infinitesimal change
in the value of the function f(x)
resulting from applying its rate of change f’(x)
over a tiny change in input dx

we may take a derivative of any function
we may define a function
as the derivative of any function
what are derivative functions
of derivative functions?

Derivation

to return the derivative

 many rules have been found
in patient study
of how functions shift
as input shifts infinitesimally
in increments of dx

let f(x) be any function
let f’(x) be the derivative of that function

for f(x) = c where c is a constant

its curve is a flat line

its rule does not vary

change is absent

its slope is zero

we write d/dx f(c)
we say dee dee x of f of c
we mean the derivative of f(c)
we mean its slope is 0
we mean its rate of change is zero

d/dx f(c) = 0 where c is a constant
f’(c) = 0 where c is a constant

because c does not vary
its slope is zero
its rise is flat
its change is none

constants
fall away in derivation
they are left behind
they zero out

we call this the constant rule

for f(x) = ax + b
its curve is a straight line
change is steady
its rule is steady

a is the slope at x
a is the rate of change at x
a is the derivative for any x

d/dx f(x) = a
f’(x) = a

we call this the linear rule

what is the derivative
if a = 0?

for f(x) = kxⁿ
change quickens
its rule quickens
its curves quicken

knxⁿ⁻¹ is the slope at x

knxⁿ⁻¹ is the rate of change at x
knxⁿ⁻¹ is the derivative for any x

d/dx f(x) = knxⁿ⁻¹

f’(x) = knxⁿ⁻¹

we call this the power rule

what is the derivative when n = 1?

these are the bones of motion

what are its sinews?
what are its tooth and nail?

what is its flesh?

follow the function
follow change
follow the line
follow its slope

where the curve turns
where change lives
the derivative
tells all

Differentiation

change is difference
between instants

a curve is the plot of a function f(x)
a curve is the plot of instants
as one thing varies with another
as y varies with x
f(x) = y

change is the difference in y
between one instant and another
separated by an interval
by a change in x

we call differentiation
the process of breaking a curve f(x)
into differences
into differentials
into derivatives

the primary result of differentiation
is f’(x) = d/dx f(x)

how does a functions output change
given an infinitesimal change
in its input?



The Integral

let f(x) be any function

f(x) dx
means a product
of height and width
an infinitesimal rectangle
f(x) by dx
an area
an amount
beneath a point on the curve

suppose we wish
to know the whole
from the parts

given f(x)
given the slivers of f(x)
given its differential f’(x)
how make them whole?

we sum all of these differentials
each too small to matter
yet together
they restore integrity
they restore f(x)

we call the integral of f(x) dx
the indefinite integral of f(x)
the sum of areas beneath f(x)
the sum of amounts beneath f(x)
f(x) made whole

we write ∫ f(x) dx
we say the integral of f of x dee x
we say the integral of f(x)
we say the antiderivative of f(x)
we mean the indefinite integral of f(x) dx
we mean sum the differentials of f(x)
we mean integrate the differentials of f(x)
we mean the accumulated quantity of f(x) as of x

we define a new function
we write F(x) = ∫ f(x) dx
we say F of x equals the integral of f(x) dx
we mean F(x) is the indefinite integral of f(x) dx
we mean F(x) is the function whose derivative is f(x)
we mean F(x) is the function whose rate of change is f(x)
we mean F(x) is the area accumulated as of x
we mean F(x) is the amount accumulated as of x

d/dx F(x) = f(x)

∫ f(x) dx = F(x) + C
where C is the constant of integration

the constant of integration C
a placeholder in memory of
that constant lost in derivation
its presence is certain
its value unknown

yet given any pairing of f(x)
(x, f(x)) or (x, y)
we may solve for C
and so find its value
completing the whole
that integration began

we define a new function
we write F′(x) = d/dx F(x)
we say F prime of x equals dee dee x of F of x
we mean F′(x) is the derivative of F(x)
we mean F′(x) is the rate of change of F(x)
we mean F′(x) is the slope of F(x)
we mean F′(x) tells how F(x) changes
  as x changes infinitesimally

suppose we wish
to know a portion
of the parts

given a range of slivers of f(x)
given a range of differentials of f(x)
how know their value A?

we sum a range of differentials
each too small to matter
yet together
they have amount A
they have area A

we write ∫ₐᵇ f(x) dx
we say the integral from a to b of f of x dee x
we mean a definite integral of f(x) dx
we mean the sum of differentials of f(x)
we mean integrate f(x) dx from x=a to x=b
we mean the area A under f(x) dx from x=a to x=b
we mean the amount A accumulated under f(x) d x from x=a to x=b

∫ₐᵇ f(x) dx = A


***

a waving hand
a falling leaf
a story told in motion

we add up slivers
to restore a whole

Integration

to integrate a function

many rules have been found

by patient study

of how functions accumulate

as input shifts

by infinitesimal steps of dx

let f(x) be any function

***

The Constant Rule

for f(x) = c
where c is a constant
change is absent
its curve is a flat line
the output is steady

the area grows linear

∫ c dx = c·x + C

the integral of a constant
is the constant times x
plus the constant of integration
that placeholder for
what derivation lost

***

The Power Rule

for f(x) = xⁿ
where n ≠ –1
increase the exponent by one

raise x to that power

divide by the new exponent

∫ xⁿ dx = (xⁿ⁺¹) / (n+1) + C

the power increases
the rule follows
the constant completes

***

The Log Rule

for f(x) = 1/x
its value and slope
diverge explosively
as x approaches zero
without ever touching

the integral is a logarithm
of x made positive
plus the constant of integration

∫ (1/x) dx = ln|x| + C

the area grows logarithmically
accumulating
at rate f(x)
the constant completes

***

The Exponential Rule

for f(x) = eˣ

its rate of change is itself
the slope is the height
the height is the slope
the integral is eˣ
plus the constant of integration

∫ eˣ dx = eˣ + C
no change in form
just a change in frame
the constant completes

***

Linearity

for f(x) = a·g(x) + b·h(x)

when a function is expressed

 as the sum of two scaled functions

 integration respects addition
integration respects scaling

∫ (a·g(x) + b·h(x)) dx  

= a·∫ g(x) dx + b·∫ h(x) dx  

= a·G(x) + b·H(x) + C

add or scale

before or after
the sum is preserved
the constant completes

***

these are the flesh
built on the
bones of motion

follow the function
follow its changing
sum its slivers
restore the curve

where change accumulates

where the curve returns

the integral

tells all

Duality

two mirrors reflect

one truth

vanishing into depth

slicing change
into infinitesimal differentials
we name differentiation 

summation of these slivers
restoring a whole
we name integration

the whole is differentiated
the differials are integrated
the whole is parted
the parts are the whole

together
they are a curve
a surface
a world

how can so little
reflect so much?

d/dx ∫ₐˣ f(t) dt = f(x)

---

some change is steady
like the moons traverse
across the sky

some change shifts
like gusts of wind
sharp and sudden

begin with that which changes steady
begin with that which changes smooth

find your way to ride
the inconstant wind