23 October 2001

Notes Prepared by Porter Johnson

**Don Kanner (Lane Tech HS, Physics) Inertia
Don** handed out selected portions of the authorized English
translation of landmark book

- The quantity of matter is the measure of the same, arising from its density and bulk conjointly.
- The quantity of motion is the measure of the same, arising from the velocity and quantity of matter conjointly.
- The
**vis insita**, or innate force of matter, is a power of resisting, by which every body, as much as in it lies, continues in its present state, whether it be of rest, or of moving uniformly forwards in a right line. - An impressed force is an action exerted upon a body, in order to change its state, either or rest, or of uniform motion in a right line.
- A centripetal force is that by which bodies are drawn or impelled, or any way tend, towards a point as to a centre.

- Every body continues in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed upon it.
- The change of motion is proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.
- To every action there is always opposed an equal reaction; or, the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.

Curiously, Newton distinguishes an innate force of matter
(inertia?) in **#3** from an impressed (applied) force in **#4**.

**Porter Johnson** commented that the question of whether Newton
actually discovered his laws by himself has been hotly debated over the
years. Consider this excerpt from a Newton Biography: http://www.ing.iac.es/PR/int_info/intisaac.html

Isaac Newton was born at Wolsthorpe, Lincolnshire on 25 December 1642. Born into a farming family and first educated at Grantham, Isaac Newton was sent to Trinity College, Cambridge, where as an undergraduate, he came under the influence of Cartesian philosophy. When confined to his home at Woolsthorpe by the plague between 1665 and 1666 Newton carried through work in the analysis of the physical world which has profoundly influenced the whole of modern science.

On returning to Cambridge, Newton became a Fellow of Trinity College, and was then appointed to the Lucasian Chair of mathematics in succession to Isaac Barrow. In the 1670s lectures, demonstrations and theoretical investigations in optics occupied Newton. In 1672 he constructed the reflecting telescope today named after him, but in the early years of the 1680s correspondence with Robert Hooke re-awakened his interest in dynamics. After Edmond Halley's visit to Cambridge to encourage him in this work, Newton laid the foundations of classical mechanics in the composition of his fundamental work Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy), which was presented to the Royal Society in 1686, and its subsequent publication being paid for by his close friend Edmund Halley.

Consider also this excerpt from the Biography of Robert Hooke: http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Hooke.html

In 1672 Hooke attempted to prove that the Earth moves in an ellipse round the Sun and six years later proposed the inverse square law of gravitation to explain planetary motions. Hooke wrote to Newton in 1679 asking for his opinion:-... of compounding the celestiall motions of the planetts of a direct motion by the tangent (inertial motion) and an attractive motion towards the centrall body ... my supposition is that the Attraction always is in a duplicate proportion to the Distance from the Center Reciprocall...

Hooke seemed unable to give a mathematical proof of his conjectures. However he claimed priority over the inverse square law and this led to a bitter dispute with Newton who, as a consequence, removed all references to Hooke from the Principia.

For balance, look at the corresponding Newton Biography on the St Andrews website: http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Newton.html.

Consider also the excerpt from this source: http://physics.ucsc.edu/~michael/

Michael Nauenberg, University of California, Santa Cruz, who organized the session, presented a paper entitled, "Newton's Early Computational Method for Dynamics." He began by observing that despite considerable historical research, very little is known about how Newton developed the mathematical theory of orbital dynamics which culminated in the Principia. A letter from Newton to Hooke, written on Dec. 13, 1679, reveals that Newton had made considerable more progress in understanding central force motion than had been previously realized. In particular a careful analysis of the original diagram which appears in this letter reveals that by then Newton understood by the fundamental symmetries of orbital motion for central forces. Moreover, the text of the letter indicates that he had developed a computational method to evaluate orbital motion for arbitrary central forces. Nauenberg went on to show that the early mathematical method Newton used to solve orbital motion for general central forces in his letter to Hooke was based on the calculus of curvature which he developed in the late 1660's. In correspondence with Newton in late 1679, Hooke suggested an alternative physical approach to which Newton gave a mathematical formulation without acknowledging Hooke (later in 1686 he wrote to Halley emphatically denying that Hooke had made any important contributions). This approach led Newton immediately to the discovery of the physical basis of Kepler's area law, which remained hidden in his earlier curvature method. The new approach is described in Proposition I, Theorem I of the Principia, and constitutes the cornerstone for the geometric methods in the book.

**Bill Colson (Morgan Park HS, Mathematics) Kitty Ball
Bill** obtained the hard plastic ball from

**Monica Seelman (ST James School) Casting out Nines
Monica** pointed out that you can check arithmetical operations by
calculating the entries modulo base 9, and then checking the
arithmetical operations modulo 9. This check on arithmetic would
work on any base, but it is especially convenient using modulo 9, since
you get the number mod 9 by repeatedly summing the digits. For
example,

Addition:
Original Problem Modulo 9
362 --->
2 |
Subtraction: Original Problem Modulo 9
5273 --->
8 |
Multiplication: Original Problem Modulo 9
635 --->
5 |

**Fred Schaal **mentioned that in hexadecimal notation, in which
the counting sequence is

**1 2 3 4 5 6
7 8 9 A B C D E F 10 ...**

he just turned the age of **3F**, and next year would become age
**40**. Dream on about hex code and remember what the Beatles
said [http://www2.uol.com.br/cante/lyrics/Beatles_-_When_I_am_64.htm],
**Fred!**

**Fred Schaal (Lane Tech HS, Mathematics) 9 ´
9 Magic Squares
**Fred handed out a sheet containing the following empty lattice:

**Fred** then asked for a **start number** (we chose **11**),
as well as an **add number** (we chose **17**). Next, he
put **11** into the middle square on the top row. The idea is
to implement "toroidal topology" with periodic boundary conditions, and
to add **17** sequentially to the **11**, and the total placed
one square above and to the right of the previous element. The
first few numbers are shown below:

11 | ||||||||

147 |
||||||||

130 | ||||||||

113 | ||||||||

96 | ||||||||

79 | ||||||||

62 | ||||||||

45 | ||||||||

28 |

At this point we do not put **164 (147 + 17)** into the location
already occupied by **11**; instead we put the **164** under
the **147**, and continue until we hit the next snag:

11 | 198 | |||||||

147 | 181 | |||||||

130 | 164 | |||||||

113 | 300 |
|||||||

96 | 283 | |||||||

266 | 79 | |||||||

62 | 249 | |||||||

45 | 232 | |||||||

28 | 215 |

once again, we proceed by putting the next number, **317**,
under the **300**, and continue the procedure. to get

793 | 980 | 1167 | 1354 | 11 | 198 | 385 | 572 | 759 |

963 | 1150 | 1337 | 147 | 181 | 368 | 555 | 742 | 776 |

1133 | 1320 | 130 | 164 | 351 | 538 | 725 | 912 | 946 |

1303 | 113 | 300 | 334 | 521 | 708 | 895 | 929 | 1116 |

96 | 283 | 317 | 504 | 691 |
878 | 1065 | 1099 | 1286 |

266 | 453 | 487 | 674 | 861 | 1048 | 1082 | 1269 | 79 |

436 | 470 | 657 | 844 | 1031 | 1218 | 1252 | 62 | 249 |

606 | 640 | 817 | 1014 | 1201 | 1235 | 45 | 232 | 419 |

623 | 810 | 997 | 1184 | 1371 | 28 | 215 | 402 | 589 |

The sum of every row, every column, every diagonal, and every set of
numbers symmetrically placed about the diagonal is **6219**, which
is **9** multiplied by the central element, **691**. Why?

**Porter Johnson** suggested subtracting the **start number 11**
from each element, and then dividing each element by the **add number
17**, to obtain the following array:

46 | 57 | 68 | 79 | 0 | 11 | 22 | 33 | 44 |

56 | 67 | 78 | 8 | 10 | 21 | 32 | 43 | 45 |

66 | 77 | 7 | 9 | 20 | 31 | 42 | 53 | 55 |

76 | 6 | 17 | 19 | 30 | 41 | 52 | 54 | 65 |

5 |
16 |
18 |
29 |
40 |
51 |
62 |
64 |
75 |

15 | 26 | 28 | 3 | 50 | 61 | 63 | 74 | 4 |

25 | 27 | 38 | 49 | 60 | 71 | 73 | 3 | 14 |

35 | 37 | 48 | 59 | 70 | 72 | 2 | 13 | 24 |

36 | 47 | 58 | 69 | 80 | 1 | 12 | 23 | 34 |

The sums are equal to **9** multiplied by the central number **40**,
or **360**. This property remains valid if you make the
following operations:

- Reflect the matrix about the middle horizontal row (
**elements in bold**). - Reflect the matrix about the middle vertical row.
- Reflect the matrix about either diagonal.

Very interesting, **Fred**!

**Ann Brandon**** (Joliet West HS, Physics) Pressure
**

As an additional application of air pressure, she showed a pair of **dent
pullers**, available at local hardware stores for about **$1**.
Dent pullers work better, and they cost less than the **Magdeberg
Hemispheres** available at science supply houses http://www.sciencekit.com/store/catalog/product.jsp?product_id=8879857.

Ann next showed the **Bed of Nails Demo**, showing the effects
of a uniform force distributed over multiple points, and then only at
one point. This apparatus, shown below, is available from the following
Educational Supply house:

Tonawanda Products Inc.

653 Erie Ave

N Tonawanda, NY 14120

Phone: 716-743-2021

Fax: 716-743-2787

She blew up a balloon, and placed it under a platform held in place on a bed of nails. Then, she placed weights on top of the platform, until the balloon burst. She then repeated the experiment, using only one nail instead of the bed of nails.

**Marilynn Stone (Lane Tech HS, Physics) Home Made System to
Illustrate Circuits**

**Marilynn** gave us the following diagram for her circuit from the
last meeting, 09 October 2001, with
directions:

Click here for a larger image.

Notes taken by **Porter Johnson**