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OpenGL Simple skinning example?

5 posts in this topic

I'm using the latest version of OpenGL and SDL 2.0. Here is my current rendering code to render an object:

    for(a=0;a<obj->total_tri;a++) {
if(!a || obj->tri[a].tex!=obj->tri[a-1].tex) {
glBindTexture(GL_TEXTURE_2D,texture[obj->tri[a].tex]);
glEnable(GL_TEXTURE_2D);
}
glBegin(GL_TRIANGLES);
glTexCoord2f(obj->vt[obj->tri[a].vt1].u,obj->vt[obj->tri[a].vt1].v);
glNormal3f(obj->vn[obj->tri[a].vn1].x,obj->vn[obj->tri[a].vn1].y,obj->vn[obj->tri[a].vn1].z);
glVertex3f(obj->v[obj->tri[a].v1].x,obj->v[obj->tri[a].v1].y,obj->v[obj->tri[a].v1].z);

glTexCoord2f(obj->vt[obj->tri[a].vt2].u,obj->vt[obj->tri[a].vt2].v);
glNormal3f(obj->vn[obj->tri[a].vn2].x,obj->vn[obj->tri[a].vn2].y,obj->vn[obj->tri[a].vn2].z);
glVertex3f(obj->v[obj->tri[a].v2].x,obj->v[obj->tri[a].v2].y,obj->v[obj->tri[a].v2].z);

glTexCoord2f(obj->vt[obj->tri[a].vt3].u,obj->vt[obj->tri[a].vt3].v);
glNormal3f(obj->vn[obj->tri[a].vn3].x,obj->vn[obj->tri[a].vn3].y,obj->vn[obj->tri[a].vn3].z);
glVertex3f(obj->v[obj->tri[a].v3].x,obj->v[obj->tri[a].v3].y,obj->v[obj->tri[a].v3].z);
glEnd();
}

This renders objects correctly, but I am going to need to implement vertex buffer objects or whatever the most recent thing is. However, I will also need to implement skeletal animation.

All of the tutorials that I find do not do anything at a low level in C, and they make use of various libraries. Since I am using a custom format, though, these libraries will not be of much use.

Can anyone link to a simple tutorial that can explain this? What must be done is simple. There are two bones. The first bone rotates the entire object, while the second bone rotates half of the object. Skinning is to be used so some vertices from the second object stay at the end of the first object, like a snake.

A simple example using an object composed of two rectangles and with all of the values hard-coded will do. Thanks!

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I would say don't bother with skeletal animation for now and just convert what you have already to using VBO's, as this could be a large chunk of work.

Once you have your code in a stable condition using the new approach then start to look at implementing skeletal animation.

JB

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Thanks for the reply! I have been looking into VBOs, but I can't seem to incorporate this code without a crash occurring: http://www.opengl.org/wiki/VBO_-_just_examples

For now, it loads .obj files into a format that can be used to export to any format, so loading into a VBO would be simple if I could find a good VBO example. What I linked to seems to be simple, but it is crashing for me on the "glDrawElements" line. Most of the VBO examples I find are either incomplete code, or ten different header files that utilize over five libraries.

Do you have any links to concise code like this?: http://openglsamples.sourceforge.net/triangle.html

Thanks!

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Personally I used http://open.gl/drawing . The tutorials and examples are very well set out and explained thoroughly.

If you are struggling to get it implemented with your current code then create a much simpler case and maybe just output some predefined triangles as shown on the website.

hope this helps

JB

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If you are using glVertex3f and friends, then you are *not* using the latest version of OpenGL.  You are, in fact, using legacy OpenGL.  Jordan already beat me to it (darn you sir) and I do recommend his link.

If you prefer to use legacy OpenGL (2.1) over core OpenGL (3.0+), then okay.  Occasionally, there may be a good reason for using legacy OpenGL.  For my latest title, I used legacy OpenGL to keep portability to OpenGL ES 2.0 easy (but it was still a mistake I regret).

If you need a tutorial on bone animation, check out this these:

http://www.codesampler.com/oglsrc/oglsrc_11.htm#ogl_skinning

http://content.gpwiki.org/index.php/OpenGL:Tutorials:Basic_Bones_System

http://prideout.net/blog/?p=56

Hope this helps,

Shogun.

Edited by blueshogun96
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Thanks! These are very useful! I actually wrote an animation tool a while back that utilized CPU skinning, which was slow and glitchy, so I'm mostly just looking for skinning examples. This (http://www.codesampler.com/oglsrc/oglsrc_11.htm#ogl_skinning) is exactly what I was looking for recently, thanks! I found it a long time ago, but I couldn't seem to find it again until you linked to it. Thanks!

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https://github.com/sp4cerat/Planet-LOD/blob/master/src.simple/Main.cpp
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Toastmastern

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• By mapra99
Hello

I am working on a recent project and I have been learning how to code in C# using OpenGL libraries for some graphics. I have achieved some quite interesting things using TAO Framework writing in Console Applications, creating a GLUT Window. But my problem now is that I need to incorporate the Graphics in a Windows Form so I can relate the objects that I render with some .NET Controls.

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• Hello, I have been working on SH Irradiance map rendering, and I have been using a GLSL pixel shader to render SH irradiance to 2D irradiance maps for my static objects. I already have it working with 9 3D textures so far for the first 9 SH functions.
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• By KarimIO
EDIT: I thought this was restricted to Attribute-Created GL contexts, but it isn't, so I rewrote the post.
Hey guys, whenever I call SwapBuffers(hDC), I get a crash, and I get a "Too many posts were made to a semaphore." from Windows as I call SwapBuffers. What could be the cause of this?
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static PIXELFORMATDESCRIPTOR pfd = // pfd Tells Windows How We Want Things To Be { sizeof(PIXELFORMATDESCRIPTOR), // Size Of This Pixel Format Descriptor 1, // Version Number PFD_DRAW_TO_WINDOW | // Format Must Support Window PFD_SUPPORT_OPENGL | // Format Must Support OpenGL PFD_DOUBLEBUFFER, // Must Support Double Buffering PFD_TYPE_RGBA, // Request An RGBA Format 32, // Select Our Color Depth 0, 0, 0, 0, 0, 0, // Color Bits Ignored 0, // No Alpha Buffer 0, // Shift Bit Ignored 0, // No Accumulation Buffer 0, 0, 0, 0, // Accumulation Bits Ignored 24, // 24Bit Z-Buffer (Depth Buffer) 0, // No Stencil Buffer 0, // No Auxiliary Buffer PFD_MAIN_PLANE, // Main Drawing Layer 0, // Reserved 0, 0, 0 // Layer Masks Ignored }; if (!(hDC = GetDC(windowHandle))) return false; unsigned int PixelFormat; if (!(PixelFormat = ChoosePixelFormat(hDC, &pfd))) return false; if (!SetPixelFormat(hDC, PixelFormat, &pfd)) return false; hRC = wglCreateContext(hDC); if (!hRC) { std::cout << "wglCreateContext Failed!\n"; return false; } if (wglMakeCurrent(hDC, hRC) == NULL) { std::cout << "Make Context Current Second Failed!\n"; return false; } ... // OGL Buffer Initialization glClear(GL_DEPTH_BUFFER_BIT | GL_COLOR_BUFFER_BIT); glBindVertexArray(vao); glUseProgram(myprogram); glDrawElements(GL_TRIANGLES, indexCount, GL_UNSIGNED_SHORT, (void *)indexStart); SwapBuffers(GetDC(window_handle));

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2019
Том 71
№ 11

# <NAME>.

Articles: 1
Brief Communications (English)

### Coordinated approximation method for nonlinear ill-posed problems

Ukr. <NAME>. - 1994. - 46, № 7. - pp. 956-961

A generalization of the method of coordinated approximation suggested by <NAME> [1] for the space $L_2(0, 1)$ is developed for abstract Hilbeit spaces. In particular, it is shown that, for $L_2(0, 1)$, some assumptions concerning ал exact solution can be weaken.


text:
# [SOLVED]Group Rings

#### Cbarker1

##### Active member
Dear Everyone,

Does every theorem that holds for finite group holds for ring groups? Why or Why not?

Thanks
Cbarker1

#### Olinguito

##### Well-known member
What do you mean by “ring groups”?

#### Cbarker1

##### Active member
A group ring defined as the following from Dummit and Foote:

Fix a commutative ring $R$ with identity $1\ne0$ and let $G=\{g_{1},g_{2},g_{3},...,g_{n}\}$ be any finite group with group operation written multiplicatively. A group ring, $RG$, of $G$ with coefficients in $R$ to be the set of all formal sum

$a_1g_1+a_2g_2+\cdots+a_ng_n$, $a_i\in R$, $1\le i\le n$

#### Olinguito

##### Well-known member
Well, note that $G$ is a group whereas $RG$ is a ring, so not every theorem about $G$ may be applicable to $RG$. For example, $G$ may be a cyclic group, but there is no such thing as a cyclic ring, so a theorem about cyclic groups may not make sense when applied to rings.

What you can say is that $RG$ contains a subring isomorphic to $R$, namely
$$\{a\cdot e_G:a\in R\}$$
as well as a subset which, with respect to multiplication, forms a group isomorphic to $G$, namely
$$\{1_R\cdot g:g\in G\}.$$


text:
# Lava Flow Mine

The official world for Lava Flow Mine is world 71 (P2P).

The Lava Flow Mine is one of the most important locations of Keldagrim, being the main source of the dwarven city's power supply. It is run collectively by the Consortium and overseen by <NAME>. The mine was first released along with the quest King of the Dwarves on 4 January 2011. Later, a training area associated with the mine was released on 16 May 2011. It is a place for players to train Mining quickly without gaining ores (though gems may be acquired) and by pleasing the Liquid Gold Nymph, players may attain pieces of the golden mining suit. A Mining level of 65, a Smithing level of 70, and a Strength level of 77 are required to complete the quest and are therefore required to use the mine.

The Lava Flow Mine acts as a geothermal power plant, using Gielinor's thermal energy (the lava) to turn water into steam, which in turn operates the various mechanisms around Keldagrim.

Mining urns, Bonus experience, and clan avatar experience boosts work at the Lava Flow Mine.

## History

A statue in the Lava Flow Mine.

Located in Keldagrim, the Lava Flow Mine is used as the primary power source for the city. It creates steam, on which many of the city's machines work (similar to the Blast Furnace). It is a vital part of Keldagrim - the Green Gemstone Director calls it the single most important piece of machinery in Keldagrim.

At the start of King of the Dwarves, a cloaked figure blows himself up in the mine, causing a cave-in, which wounds six of the miners. He blows up the secondary boilers only, and black guard troops come to fix them very soon. They leave the miners under the rubble, for the player to mine them out. Though at first it seemed that the guards were not concerned about the trapped miners, it is revealed that if the boilers weren't fixed, they might have exploded, causing further deaths and leaving most of Keldagrim without power for days. After the quest it seems 3 of the 6 miners have died: 2 died immediately, and 1 more died of his wounds later on.

During the quest the Blue Opal Director reveals how the lava flow mine works. The lava comes from the sub-Barendir magma vent. It is siphoned off from this vent and run along channels in the mine. The lava heats water in primary boilers over the channels, and the steam created is channelled into seven secondary boilers in the centre of the mine. From here, the steam is pushed through distribution vents to the various machines around Keldagrim, such as the mine carts and the blast furnace.

## Location

The Lava Flow Mine is located through an opening on the east wall of Keldagrim. The easiest way to get there is by using the mine cart from the Grand Exchange to travel to East Keldagrim, and then run northeast.

## Requisites

### Recommended

Optional: Ghostspeak amulet, cramulet, or completion of the hard Morytania achievements to speak to the Dwarven ancestor spirit. This is required to understand him and hear his story. These items are not needed to get the reward +10% experience for 10 minutes.

Segment Location Table

## Mining

The main purpose of the Lava Flow Mine is to mine off crust from one of the four lava channels. Foreman Jaak will say which of the four lava channels needs the most attention, which will also yield the most mining experience per successful cut of the crust. You can check the steam gauges next to each channel segment to see the flow in that segment - the segment with 50% flow gives the most experience.

After a while (Anywhere between 4 to 7 minutes), the best flow changes, you will receive a message in your chatbox when this happens. Jaak will know which one yields the most experience again. Unless the world is fairly full, it is advised to talk to Jaak instead of finding the correct flow yourself, because finding it with too few people might take too long. The segments next to the one with 50% flow will have 60% flow, and the next segments 70% flow and so on. This can help in finding the best segment to mine.

When mining the best channel segment, one successful cut yields 100 experience when the exact correct part of the correct flow is mined. Experience gained is proportional to the flow in that segment, as shown in the table below.

Flow rate Experience per mine
50% 100
60% 90
70% 80
80% 70
90% 60
100% 50

Strange rocks can be obtained from mining the crusts.

### Finding the blockage

When the blockage segment moves, immediately go to Foreman Jaak near the entrance to find the new channel. Once you know this information, go to the steam gauge for either segment 4 or 5 of the new channel and find the percentage. If it isn't 90% (which only one end can be from the centre segments) or 50%, then check the other segment's value and use that as an offset to find the location.

Segment 5 Segment 4 Blockage
90% 80% 1
80% 70% 2
70% 60% 3
60% 50% 4
50% 60% 5
60% 70% 6
70% 80% 7
80% 90% 8

As an alternative to asking Jaak, the correct channel can be found by reading the flow at two central locations in segments 4 or 5: between channels A and B, and between channels C and D. At each location read the two dials, and stop as soon as you find the one with less than 100%. Then read the neighbouring dial to find the direction of the correct segment.

### Maximising experience rates

To maximise experience rates while in the Lava Flow Mine, you must be mining the segment with 50% flow. This can be quickly identified by firstly looking in segments 4 or 5 of each channel to find the one with below 100% flow, or talking to <NAME> to be told which channel letter is slower, and then moving along it until you find the 50% flow, once here, you should begin mining.

Also, it is recommended to mine in any of the unofficial worlds with a clan to find the 50% segments faster.

At level 80, the maximum experience per hour with only a lava titan, no other boosting equipment/items, and a pickaxe:

Pickaxe XP per hour (no urns) XP per hour (with decorated urns) XP per hour (with decorated urns + urn enhancer)
• Necronium pickaxe + 4
• Crystal pickaxe
60000 72000 75000
• Bane pickaxe
62000 74000 78000
• Bane pickaxe + 4
• Imcando pickaxe
68000 80000 85000

The current maximum experience is around 100,000 per hour (200,000 per hour with Bonus experience) at level 99 with a good team and a lava titan.

## Distractions

There are multiple distractions that can occur while mining the crusts.

  template = :Lava Flow Mine/Boiler calculator/Temp form = CalcForm result = CalcResult param = Smithing|Smithing level (70-90)|70|int|70-90  This text will disappear if the form is loaded properly.
• Players fixing a broken boiler in the Lava Flow Mine.
Boiler breakages - As a result of the chaos dwarf attack, the boilers overheat. Repairing them takes a random Smithing level (70 to 90, boosts may be used) and a hammer (tool belt works), which spawns at the mine entrance. If you fix it you will gain Smithing experience based on the required level. This cannot be assisted. A boiler will break on average about every 1 hour and 10 minutes[source needed], although the time between breakages is random. You will get a red message in your chatbox whenever a boiler breaks.
• To see how much experience you would get when repairing a boiler, see the calculator to the right. You can only repair one boiler per hour.
• This formula is ${\displaystyle 150\times L-7500}$ where ${\displaystyle L}$ is the smithing level required.
• TzHaar hero - The TzHaar hero will chase a lava monster along the channels of the mine. Players can stall the lava monster by taunting it. If they can do so for long enough for the TzHaar hero will catch it, everyone inside the mine will be twice as likely to find gems for the next ten minutes. You will get a red message in your chatbox when the bonus comes into effect and also when it ends.
• Possessed pickaxe - A Dwarven ancestor spirit will take control of a nearby pickaxe. If the pickaxe is defeated, the ancestor will grant a 10% Mining experience bonus for ten minutes to the player who killed it. You will get a red message in your chatbox when the bonus comes into effect and also when it ends. The bonus stacks multiplicatively with a Bonus XP Weekend multiplier. Note that killing multiple Possessed pickaxes and speaking to the Dwarven ancestor spirit within 10 minutes of one another will not add an additional 10% bonus but will restart your 10 minutes.
• Liquid Gold Nymph - The Liquid Gold Nymph appears once you fill your progress bar. She will give a piece of the golden mining suit when talked to. If you already have a full set of the outfit, she will offer you 2,500 Mining experience or the option to gild any single pickaxe you have in your inventory or on the tool belt (not including the inferno adze, crystal pickaxe, dwarven army axe, or sacred clay pickaxe). You will get a message in your chatbox when she appears, and a sound will play. The progress bar (Upper, blue bar) fill rate is affected by the stamina bar (lower, yellow bar), the possible progress you do per strike is based on whether the stamina bar is 100%, 1-99%, or 0% full, doing 100%, 90%, or 20% damage respectively. So keeping it from becoming empty by clicking on the mining location again at any time before or after it becomes empty makes this process faster. It IS possible to four tick this by clicking every 2.4 seconds, but as this only increases your damage from 90% to 100%, it’s likely not worth the effort. Each bar takes approximately 10-20 minutes to fill depending on stamina bar maintenance.
• Lava geyser - (requires completion of Birthright of the Dwarves): Occasionally a lava geyser will appear in one of the lava sections. First, players have to cool it down by using Water or Ice spells. After it has cooled down, it can be mined for a hefty amount of Mining experience. When the cooled down geyser has been mined completely, it will disappear and players will receive an Imcando pickaxe fragment.
• Note: It is announced in the chatbox when a geyser appears.

## Rewards

### Golden mining suit

In addition to the Mining and Smithing experience obtained by contributing to the mine, the Liquid Gold Nymph will appear when the progress bar is filled and reward players with a piece of the golden mining suit in the following order. Wearing all five pieces gives a 6% boost to experience while Mining (1% set bonus), similar to the effects of the lumberjack clothing set. It is no longer possible for a player to obtain multiple suit pieces. Previous to a recent update, one could drop a piece that they wish to obtain and then talk to the nymph for another then proceed to pick up the piece they dropped after they receive the piece they want from the nymph (The option to drop was replaced with destroy).

Image Item Boost
Golden mining boots 1%
Golden mining gloves 1%
Golden mining helmet 1%
Golden mining trousers 1%
Golden mining top 1%*
Full set 6.0%

*If the player owns any Varrock armour, the mining top will also give the chance of receiving extra ores (up to whatever ore your Varrock armour would normally allow).

### Gilded Pickaxes

The Liquid Gold Nymph will also offer to trim the pickaxe in the players inventory or tool belt, if they have acquired the complete golden mining suit. Foreman Jaak can remove the gilding on any of the pickaxes.

The player also has the option to get a 2500 experience reward instead of a gilded pickaxe, by asking if the Liquid golden nymph has nothing other than gold.

Image Item
Gilded bronze pickaxe
Gilded iron pickaxe
Gilded steel pickaxe
Gilded mithril pickaxe
Gilded rune pickaxe
Gilded dragon pickaxe

### Imcando pickaxe

The imcando pickaxe.

During the Lava geyser distraction, the player has the chance to obtain Imcando pickaxe fragments. These fragments can be used to create an Imcando pickaxe. To do this the player will need to bring the following to Thurgo after completing Birthright of the Dwarves:

## Update history

The update history project is a work-in-progress – not all updates to this topic may be covered below. See here for how to help out!
• patch 14 January 2019 (Update):
• Fixed an issue where multiple Liquid Gold Nymphs could be spawned under certain conditions.
• patch 7 January 2019 (Update):
• The nymph in the lava flow mine now spawns more frequently and consistently.
• patch 26 August 2014 (Update):
• The chi-blast mining override now plays correctly in the Lava Flow Mine.
• patch 8 October 2013 (Update):
• A debug message that appeared whilst mining from a lava geyser in the Lava Flow Mine has been removed.
• patch 13 August 2013 (Update):
• Sheathed pickaxes no longer appear in two places when mining in the Lava Flow Mine.
• patch 11 January 2012 (Update):
• Portions of the bridges in the Lava Flow Mine can no longer be temporarily deleted by planting flowers.
• patch 5 July 2011 (Update):
• The Mining boost gained during Lava Flow Mine now persists over logout.
• patch 25 May 2011 (Update):
• It is now possible to gain strange rocks from the Lava Flow Mine.
• The penance horn now functions correctly in the Lava Flow Mine.

## Trivia

• If you speak with the Dwarven ancestor spirit while wearing the ghostspeak amulet or the cramulet, he will tell you a story about how he used to be the #2 miner in ancient Keldagrim, and he was killed by a lava monster before he was able to compete for the #1 miner spot.
• Mining from a lava flow or even being within the mine should kill the player due to convection and radiant heat, since the heat from the lava would convect (and be radiated) into the air making the temperature too high for a human to survive; the foreman mentions this and says it's a wonder that it doesn't happen.
• On 12 January 2012, after The Firemaker's Curse game update, players at the Lava Flow Mine found it impossible to cross the bridges to access other channels. This was later fixed in a system update.
• Sealed clue scrolls may also be obtained from mining crust.


text:
# Dielectric Sphere in Field of a Point Charge

by Apteronotus
Tags: dielectric sphere, point charge
 P: 203 Hi, I have derived the electric potential equations inside and outside the sphere due to a point charge $$q$$ placed a distance $$b$$ way from the sphere's center. The potentials are given by: $$\Phi_{in}(r,\theta) = \sum^{\infty}_{n=0} A_{n}r^{n}P_{n}(cos\theta)$$ and $$\Phi_{out}(r,\theta) = \sum^{\infty}_{n=0} \frac{kr^{n}}{b^{n+1}} + \sum^{\infty}_{n=0}\frac{B_{n}}{r^{n+1}}P_{n}(cos\theta)$$ where $$k=\frac{q}{4\pi\epsilon_{0}}$$ and $$P_{n}$$ - are the Legendre polynomials I have calculated the the constants $$A_{n}$$ and $$B_{n}$$ according to the usual boundary conditions. Unfortunately, almost non of them are equal to zero unlike the the case of a 'sphere in a uniform field'. Is there any way of truncating these infinite sums to end up with something nice and clean?


text:
# Do Math Professors still do textbook problems to learn subjects foreign to them? [closed]

This question is very subjective and is of no technical detail, however since MSE is a place for working mathematicians, I thought I'd ask here to get the best answers!

As a prospective graduate student in mathematics, I have come to know the mountain of knowledge is quite tall and there is much to learn, even past graduate school.

However, I would like to think that if my goals of becoming a professor come true and I become a specialist in an area of math, that even though I specialize in one area, that I would still love to continue learning other areas of math.

As a working mathematician in a certain area, do you ever find yourself as a working analyst picking up Hartshorne and finishing it 3 months? Do you as a differential geometer find yourself interested in Diophantine equations and pick up a graduate number theory book and go through the exercises?

If this is the case, do you normally read textbooks as a fulltime professor and go through exercises just as in university?

Or, is it unrealistic to spend time on new material when this time could be spent on your area of research?

## closed as primarily opinion-based by Peter, <NAME>, Lord Shark the Unknown, Clarinetist, Ali CaglayanSep 4 '18 at 18:35

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

• The professors I know (that are mostly from mathematical physics) usually work like this: A new area comes up in connection with a concrete question or topic of research, and they only learn the aspects relevant to their problem (that's also a matter of time.) They learn new things by applying them to their problems, and of course by talking with coworkers and colleagues. – Luke Sep 4 '18 at 12:20
• There is a graduate text that I use quite a bit which has no exercises... – <NAME> Sep 4 '18 at 12:22
• Since suprising links of different seeming topics occur very often, I would generally agree to the advice to study also other topics. But as you said in your question, it is difficult to give good avices without knowing deeply the details, and probably opinions are widely spread. – Peter Sep 4 '18 at 12:25

This is just an opinion, based on my own experience as a working mathematician. Let me say honestly that a professor seldom studies a subject as a student would do. In our times, publishing has become an urgence, so that we must publish as soon as possible.

Therefore, we do not usually move very far away from our expertise, unless we are the few outstanding mathematicians that can work on almost any subject. I could tell you a couple of names, maybe three. If, while working on something that we know well, we need to learn some new result from a neighboring area, we tend to study what we need. A textbook might be useful, but we can't read it from the beginning to the end.

Last year I had to learn some analytic number theory, and I picked up a few books on the subject. Today, I have already forgotten everything, since I did not study theorems deeply. I regret this, but learning a discipline like a student would require hours, days, weeks of hard work. I simply don't have so much time to study the way I did when I was a student.

If a mathematician needs to learn a subject and if he or she has a textbook then of course he or she does the exercises! Since he or she has presumably become a better problem solver than the typical student many of the exercises will be quite trivial, but the ones he or she doesn't get immediately are the important and useful ones.

He or she is lucky if a textbook on the subject even exists. It will often happen that there is no text - in such a case he or she, at least in my experience, will make up exercises to do. (Often without explicit thinking "hmm, what are some good exercises?" - when one learns a theorem it leads naturally to questions that one naturally tries to answer.)

I recall shortly after getting my degree: I already "knew" basic analysis, but inspired by something one of my professors said once I went through Rudin Real and Complex Analysis and did almost all of the exercises in most of the chapters. Was somewhat less stupid after doing that.

The last few years I've been learning a little (linear) algebra that I should have learned in school but didn't. I find the only things I really understand are the things I've worked out myself, so even thought plenty of books exist I've ignored them and taken things like the Jordan Canonical Form or the Cayley-Hamilton theorem as exercises... (I've decided that the key to all wisdom is the fact that $K[x]$ is a PID.)


text:
Properties

 Label 6720.cg Number of curves $4$ Conductor $6720$ CM no Rank $1$ Graph

Related objects

Show commands for: SageMath
sage: E = EllipticCurve("cg1")

sage: E.isogeny_class()

Elliptic curves in class 6720.cg

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
6720.cg1 6720y3 [0, 1, 0, -23905, 1414655] [4] 12288
6720.cg2 6720y2 [0, 1, 0, -1505, 21375] [2, 2] 6144
6720.cg3 6720y1 [0, 1, 0, -225, -897] [2] 3072 $$\Gamma_0(N)$$-optimal
6720.cg4 6720y4 [0, 1, 0, 415, 73983] [2] 12288

Rank

sage: E.rank()

The elliptic curves in class 6720.cg have rank $$1$$.

Complex multiplication

The elliptic curves in class 6720.cg do not have complex multiplication.

Modular form6720.2.a.cg

sage: E.q_eigenform(10)

$$q + q^{3} + q^{5} - q^{7} + q^{9} + 4q^{11} + 2q^{13} + q^{15} - 6q^{17} + O(q^{20})$$

Isogeny matrix

sage: E.isogeny_class().matrix()

The $$i,j$$ entry is the smallest degree of a cyclic isogeny between the $$i$$-th and $$j$$-th curve in the isogeny class, in the LMFDB numbering.

$$\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)$$

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)

The vertices are labelled with LMFDB labels.


text:
# Uniform proofs of ACC representations

Uniform proofs of ACC representations We give a uniform proof of the theorems of Yao and Beigel–Tarui representing ACC predicates as constant depth circuits with $$\hbox {MOD}_{m}$$ MOD m gates and a symmetric gate. The proof is based on a relativized, generalized form of Toda’s theorem expressed in terms of closure properties of formulas under bounded universal, existential and modular counting quantifiers. This allows the main proofs to be expressed in terms of formula classes instead of Boolean circuits. The uniform version of the Beigel–Tarui theorem is then obtained automatically via the Furst–Saxe–Sipser and Paris–Wilkie translations. As a special case, we obtain a uniform version of Razborov and Smolensky’s representation of $$\hbox {AC}^{0}[p]$$ AC 0 [ p ] circuits. The paper is partly expository, but is also motivated by the desire to recast Toda’s theorem, the Beigel–Tarui theorem, and their proofs into the language of bounded arithmetic. However, no knowledge of bounded arithmetic is needed. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Archive for Mathematical Logic Springer Journals

# Uniform proofs of ACC representations

, Volume 56 (6) – May 26, 2017
31 pages

/lp/springer_journal/uniform-proofs-of-acc-representations-z3ddXbhBsG
Publisher
Springer Berlin Heidelberg
Subject
Mathematics; Mathematical Logic and Foundations; Mathematics, general; Algebra
ISSN
0933-5846
eISSN
1432-0665
D.O.I.
10.1007/s00153-017-0560-9
Publisher site
See Article on Publisher Site

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# Isaac Newton's occult studies

Colorized engraving after Enoch Seeman's 1726 portrait of Newton

Sir <NAME> (1642–1727), the noted English scientist and mathematician, wrote many works that would now be classified as occult studies. These occult works explored chronology, alchemy, and Biblical interpretation (especially of the Apocalypse). Newton's scientific work may have been of lesser personal importance to him, as he placed emphasis on rediscovering the occult wisdom of the ancients. In this sense, some[1] believe that any reference to a "Newtonian Worldview" as being purely mechanical in nature is somewhat inaccurate.

After purchasing and studying Newton's alchemical works in 1942, economist <NAME>, for example, opined that "Newton was not the first of the age of reason, he was the last of the magicians". In the Early Modern Period of Newton's lifetime, the educated embraced a world view different from that of later centuries. Distinctions between science, superstition, and pseudoscience were still being formulated, and a devoutly Christian Biblical perspective permeated Western culture.

## Alchemical research

Much of what are known as Isaac Newton's occult studies can largely be attributed to his study of alchemy.[2] Newton was deeply interested in all forms of natural sciences and materials science, an interest which would ultimately lead to some of his better-known contributions to science. During Newton's lifetime the study of chemistry was still in its infancy, so many of his experimental studies used esoteric language and vague terminology more typically associated with alchemy and occultism.[3] It was not until several decades after Newton's death that experiments of stoichiometry under the pioneering works of Antoine Lavoisier were conducted, and analytical chemistry, with its associated nomenclature, came to resemble modern chemistry as we know it today. However, Newton's contemporary and fellow Royal Society member, <NAME>, had already discovered the basic concepts of modern chemistry and began establishing modern norms of experimental practice and communication in chemistry, information which Newton did not utilize.

Much of Newton's writing on alchemy may have been lost in a fire in his laboratory, so the true extent of his work in this area may have been larger than is currently known. Newton also suffered a nervous breakdown during his period of alchemical work, which is thought by some to have resulted from the psychological transformation alchemy was originally designed to induce[citation needed], though there is also speculation that it may have been some form of chemical poisoning (possibly from mercury, lead, or some other substance).[4]

An 1874 engraving showing a probably apocryphal account of Newton's lab fire. In the story, Newton's dog started the fire, burning 20 years of research. Newton is thought to have said,
"O Diamond, Diamond, thou little knowest the mischief thou hast done."[5]

Newton's writings suggest that one of the main goals of his alchemy may have been the discovery of The Philosopher's Stone (a material believed to turn base metals into gold), and perhaps to a lesser extent, the discovery of the highly coveted Elixir of Life.[4] Newton reportedly believed that a Diana's Tree, an alchemical demonstration producing a dendritic "growth" of silver from solution, was evidence that metals "possessed a sort of life."[6]

Some practices of alchemy were banned in England during Newton's lifetime, due in part to unscrupulous practitioners who would often promise wealthy benefactors unrealistic results in an attempt to swindle them. The English Crown, also fearing the potential devaluation of gold, should The Philosopher's Stone actually be discovered, made penalties for alchemy very severe. In some cases the punishment for unsanctioned alchemy would include the public hanging of an offender on a gilded scaffold while adorned with tinsel and other unspecified items.[4]

## Writings

Due to the threat of punishment and the potential scrutiny he feared from his peers within the scientific community, Newton may have deliberately left his work on alchemical subjects unpublished. Newton was well known as being highly sensitive to criticism, such as the numerous instances when he was criticized by <NAME>, and his admitted reluctance to publish any substantial information regarding calculus before 1693. A perfectionist by nature, Newton also refrained from publication of material that he felt was incomplete, as evident from a 38-year gap from Newton's conception of calculus in 1666 and its final full publication in 1704, which would ultimately lead to the infamous Leibniz–Newton calculus controversy. In 1936, a collection of Isaac Newton's unpublished works were auctioned by Sotheby's on behalf of <NAME>, 9th Earl of Portsmouth, who had inherited them from Newton's great-niece. Known as the "Portsmouth Papers", this material consisted of three hundred and twenty-nine lots of Newton's manuscripts, over a third of which were filled with content that appeared to be alchemical in nature. At the time of Newton's death this material was considered "unfit to publish" by Newton's estate, and consequently fell into obscurity until their somewhat sensational reemergence in 1936.[7]

At the auction many of these documents were purchased by economist <NAME>, who throughout his life collected many of Newton's alchemical writings. Much of the Keynes collection later passed to eccentric document collector <NAME>, who was himself a vigorous collector of Isaac Newton's original manuscripts.

Many of the documents collected by Keynes and Yahuda are now in the Jewish National and University Library in Jerusalem. In recent years, several projects have begun to gather, catalogue, and transcribe the fragmented collection of Newton's work on alchemical subjects and make them freely available for on-line access. Two of these are The Chymistry of Isaac Newton Project[8] supported by the U.S. National Science Foundation, and The Newton Project[9] supported by the U.K. Arts and Humanities Research Board. In addition, The Jewish National and University Library has published a number of high-quality scanned images of various Newton documents.[10]

### The Philosopher's Stone

Of the material sold during the 1936 Sotheby's auction, several documents indicate an interest by Newton in the procurement or development of the Philosopher's Stone. Most notably are documents entitled Artephius his secret Book, followed by The Epistle of Iohn Pontanus, wherein he beareth witness of ye book of Artephius; these are themselves a collection of excerpts from another work entitled <NAME>, His Exposition of the Hieroglyphicall Figures which he caused to be painted upon an Arch in St Innocents Church-yard in Paris. Together with The secret Booke of Artephius, And the Epistle of Iohn Pontanus: Containing both the Theoricke and the Practicke of the Philosophers Stone. This work may also have been referenced by Newton in its Latin version found within Lazarus Zetzner's Theatrum Chemicum, a volume often associated with the Turba Philosophorum and other early European alchemical manuscripts. <NAME>, one subject of the aforementioned work, was a notable, though mysterious figure, often associated with the discovery of the Philosopher's Stone, hieroglyphical figures, early forms of tarot, and occultism. Artephius, and his "secret book", were also subjects of interest to 17th century alchemists.

Also in the 1936 auction of Newton's collection was, The Epitome of the treasure of health written by Edwardus Generosus Anglicus innominatus who lived Anno Domini 1562. This is a twenty-eight page treatise on the Philosopher's Stone, the Animal or Angelicall Stone, the Prospective stone or magical stone of Moses, and the vegetable or the growing stone. The treatise concludes with an alchemical poem.

## Biblical studies

In a manuscript he wrote in 1704 in which he describes his attempts to extract scientific information from the Bible, Newton estimated that the world would end no earlier than 2060. In predicting this he said, "This I mention not to assert when the time of the end shall be, but to put a stop to the rash conjectures of fanciful men who are frequently predicting the time of the end, and by doing so bring the sacred prophesies into discredit as often as their predictions fail."[11]

### Newton's studies of the Temple of Solomon

Newton studied and wrote extensively upon the Temple of Solomon, dedicating an entire chapter of The Chronology of Ancient Kingdoms to his observations regarding the temple. Newton's primary source for information was the description of the structure given within 1 Kings of the Hebrew Bible, which he translated himself from Hebrew.[12]

<NAME>'s diagram of part of the Temple of Solomon, taken from Plate 1 of The Chronology of Ancient Kingdoms. Published London, 1728.

In addition to scripture, Newton also relied upon various ancient and contemporary sources while studying the temple. He believed that many ancient sources were endowed with sacred wisdom[4] and that the proportions of many of their temples were in themselves sacred. This belief would lead Newton to examine many architectural works of Hellenistic Greece, as well as Roman sources such as Vitruvius, in a search for their occult knowledge. This concept, often termed prisca sapientia (sacred wisdom and also the ancient wisdom that was revealed to Adam and Moses directly by God), was a common belief of many scholars during Newton's lifetime.[13]

A more contemporary source for Newton's studies of the temple was <NAME>, who just a few decades earlier had published an influential manuscript entitled Ezechielem Explanationes, in which Villalpando comments on the visions of the biblical prophet Ezekiel, including within this work his own interpretations and elaborate reconstructions of Solomon's Temple. In its time, Villalpando's work on the temple produced a great deal of interest throughout Europe and had a significant impact upon later architects and scholars.[14][15]

As a Bible scholar, Newton was initially interested in the sacred geometry of Solomon's Temple, such as golden sections, conic sections, spirals, orthographic projection, and other harmonious constructions, but he also believed that the dimensions and proportions represented more. He noted that the temple's measurements given in the Bible are mathematical problems, related to solutions for $\pi$ and the volume of a hemisphere, $V = (2/3)\pi r^3$, and in a larger sense that they were references to the size of the Earth and man's place and proportion to it.[citation needed]

Newton believed that the temple was designed by King Solomon with privileged eyes and divine guidance. To Newton, the geometry of the temple represented more than a mathematical blueprint, it also provided a time-frame chronology of Hebrew history.[16] It was for this reason that he included a chapter devoted to the temple within The Chronology of Ancient Kingdoms, a section which initially may seem unrelated to the historical nature of the book as a whole.

Newton felt that just as the writings of ancient philosophers, scholars, and Biblical figures contained within them unknown sacred wisdom, the same was true of their architecture. He believed that these men had hidden their knowledge in a complex code of symbolic and mathematical language that, when deciphered, would reveal an unknown knowledge of how nature works.[13]

In 1675 Newton annotated a copy of Manna - a disquisition of the nature of alchemy, an anonymous treatise which had been given to him by his fellow scholar <NAME>. In his annotation Newton reflected upon his reasons for examining Solomon's Temple by writing:

This philosophy, both speculative and active, is not only to be found in the volume of nature, but also in the sacred scriptures, as in Genesis, Job, Psalms, Isaiah and others. In the knowledge of this philosophy, God made Solomon the greatest philosopher in the world.[16]

During Newton's lifetime, there was great interest in the Temple of Solomon in Europe, due to the success of Villalpando's publications, and augmented by a vogue for detailed engravings and physical models presented in various galleries for public viewing. In 1628, Judah Leon Templo produced a model of the temple and surrounding Jerusalem, which was popular in its day. Around 1692, <NAME> produced a highly detailed model of the temple for use in an opera in Hamburg composed by <NAME>. This immense 13-foot-high (4.0 m) and 80-foot-around (24 m) model was later sold in 1725 and was exhibited in London as early as 1723, and then later temporarily installed at the London Royal Exchange from 1729–1730, where it could be viewed for half-a-crown. Sir Isaac Newton's most comprehensive work on the temple, found within The Chronology of Ancient Kingdoms, was published posthumously in 1728, only adding to the public interest in the temple.[17]

### Newton's prophecy

Newton considered himself to be one of a select group of individuals who were specially chosen by God for the task of understanding Biblical scripture.[18] He was a strong believer in prophetic interpretation of the Bible, and like many of his contemporaries in Protestant England, he developed a strong affinity and deep admiration for the teachings and works of Jose<NAME>. Though he never wrote a cohesive body of work on prophecy, Newton's belief led him to write several treatises on the subject, including an unpublished guide for prophetic interpretation entitled Rules for interpreting the words & language in Scripture. In this manuscript he details the necessary requirements for what he considered to be the proper interpretation of the Bible.

In addition, Newton would spend much of his life seeking and revealing what could be considered a Bible Code. He placed a great deal of emphasis upon the interpretation of the Book of Revelation, writing generously upon this book and authoring several manuscripts detailing his interpretations. Unlike a prophet in the true sense of the word, Newton relied upon existing Scripture to prophesy for him, believing his interpretations would set the record straight in the face of what he considered to be "so little understood".[19] In 1754, 27 years after his death, <NAME>'s treatise, An Historical Account of Two Notable Corruptions of Scripture would be published, and though it does not argue any prophetic meaning, it does exemplify what Newton considered to be just one popular misunderstanding of Scripture.

Although Newton's approach to these studies could not be considered a scientific approach, he did write as if his findings were the result of evidence-based research.

### 2060

In late February and early March 2003, a large amount of media attention circulated around the globe regarding largely unknown and unpublished documents, evidently written by <NAME>, indicating that he believed the world would end no earlier than 2060. The story garnered vast amounts of public interest and found its way onto the front page of several widely distributed newspapers, including the UK's Daily Telegraph, Canada's National Post, Israel's Maariv and Yediot Aharonot, and was also featured in an article in the scientific journal, Canadian Journal of History.[20] Television and Internet stories in the following weeks heightened the exposure and ultimately would include the production of several documentary films focused upon the topic of the 2060 prediction and some of Newton's lesser known beliefs and practices.

The two documents detailing this prediction are currently housed within the Jewish National and University Library in Jerusalem.[20] Both were believed to be written toward the end of Newton's life, circa 1705, a time frame most notably established by the use of the full title of Sir <NAME> within portions of the documents.

These documents do not appear to have been written with the intention of publication and Newton expressed a strong personal dislike for individuals who provided specific dates for the Apocalypse purely for sensational value. Furthermore, he at no time provides a specific date for the end of the world in either of these documents.[20]

To understand the reasoning behind the 2060 prediction, an understanding of Newton's theological beliefs should be taken into account, particularly his apparent antitrinitarian beliefs and his Protestant views on the Papacy. Both of these lay essential to his calculations, which ultimately would provide the 2060 time frame. See Isaac Newton's religious views for more details.

The first document, part of the Yahuda collection,[21] is a small letter slip, on the back of which is written haphazardly in Newton's hand:

Prop. 1. The 2300 prophetick days did not commence before the rise of the little horn of the He Goat.

2 Those day [sic] did not commence a[f]ter the destruction of Jerusalem & ye Temple by the Romans A.[D.] 70.

3 The time times & half a time did not commence before the year 800 in wch the Popes supremacy commenced

4 They did not commence after the re[ig]ne of Gregory the 7th. 1084

5 The 1290 days did not commence b[e]fore the year 842.

6 They did not commence after the reigne of Pope Greg. 7th. 1084

7 The diffence [sic] between the 1290 & 1335 days are a parts of the seven weeks.

Therefore the 2300 years do not end before ye year 2132 nor after 2370. The time times & half time do n[o]t end before 2060 nor after [2344] The 1290 days do not begin [this should read: end] before 2090 nor after 1374 [sic; Newton probably means 2374][20]

The second reference to the 2060 prediction can be found in a folio,[22] in which Newton writes:

So then the time times & half a time are 42 months or 1260 days or three years & an half, recconing twelve months to a yeare & 30 days to a month as was done in the Calendar of the primitive year. And the days of short lived Beasts being put for the years of lived [sic for "long lived"] kingdoms, the period of 1260 days, if dated from the complete conquest of the three kings A.C. 800, will end A.C. 2060. It may end later, but I see no reason for its ending sooner. This I mention not to assert when the time of the end shall be, but to put a stop to the rash conjectures of fancifull men who are frequently predicting the time of the end, & by doing so bring the sacred prophesies into discredit as often as their predictions fail. Christ comes as a thief in the night, & it is not for us to know the times & seasons wch God hath put into his own breast.[20]

Clearly Newton's mathematical prediction of the end of the world is one derived from his interpretation of not only scripture, but also one based upon his theological viewpoint regarding specific chronological dates and events as he saw them.

Newton may not have been referring to the post 2060 event as a destructive act resulting in the annihilation of the globe and its inhabitants, but rather one in which he believed the world, as he saw it, was to be replaced with a new one based upon a transition to an era of divinely inspired peace. In Christian and Islamic theology this concept is often referred to as The Second Coming of Jesus Christ and the establishment of The Kingdom of God on Earth. In a separate manuscript,[23] Isaac Newton paraphrases Revelation 21 and 22 and relates the post 2060 events by writing:

A new heaven & new earth. New Jerusalem comes down from heaven prepared as a Bride adorned for her husband. The marriage supper. God dwells with men wipes away all tears from their eyes, gives them of ye fountain of living water & creates all thin things new saying, It is done. The glory & felicity of the New Jerusalem is represented by a building of Gold & Gemms enlightened by the glory of God & ye Lamb & watered by ye river of Paradise on ye banks of wch grows the tree of life. Into this city the kings of the earth do bring their glory & that of the nations & the saints reign for ever & ever.[20]

## Newton's chronology

<NAME> wrote extensively upon the historical topic of Chronology. In 1728 "The Chronology of Ancient Kingdoms", an approximately 87,000 word composition that details the rise and history of various ancient kingdoms was published. The publication date of this work occurred after his death, though the majority of it had been reviewed for publication by Newton himself shortly before he died. As such, this work represents one of his last known personally reviewed publications. Sometime around 1701 he also produced a thirty page unpublished treatise entitled, "The Original of Monarchies" detailing the rise of several monarchs throughout antiquity and tracing them back to the biblical figure of Noah.[24]

Newton's chronological writing is Eurocentric, with the earliest records focusing upon Greece, Anatolia, Egypt, and the Levant. Many of Newton's dates do not correlate with current historical knowledge. While Newton mentions several pre-historical events found within The Bible, the oldest actual historical date he provides is 1125 BC. In this entry he mentions Mephres, a ruler over Upper Egypt from the territories of Syene to Heliopolis, and his successor Misphragmuthosis. However, during 1125 BC the Pharaoh of Egypt is now understood to be Ramesses IX.

Though some of the dates Newton provides for various events are inaccurate by modern standards, archaeology as a form of modern science did not exist in Newton's time. In fact, the majority of the conclusionary dates which Newton cites are based on the works of Herodotus, Pliny, Plutarch, Homer, and various other classical historians, authors, and poets; themselves often citing secondary sources and oral records of uncertain date. Newton's approach to chronology was focused upon gathering historical information from various sources found throughout antiquity and cataloguing them according to their appropriate date by his contemporary understanding, standards, and available source material.

### Newton's Atlantis

Found within The Chronology of Ancient Kingdoms, are several passages that directly mention the mythical land of Atlantis. The first such passage is part of his Short Chronical which indicates his belief that Homer's Ulysses left the island of Ogygia in 896 BC. In Greek mythology, Ogygia was home to Calypso, the daughter of Atlas (after whom Atlantis was named). Some scholars have suggested that Ogygia and Atlantis are locationally connected, or possibly the same island. From his writings it appears Newton may have shared this belief. Newton also lists Cadis or Cales as possible candidates for Ogygia, though does not cite his reasons for believing so. Within the same material Newton mentions that according to ancient sources, Atlantis had been as big as all Europe, Africa and Asia, but was sunk into the Sea.

## Newton and secret societies

<NAME> has often been associated with various secret societies and fraternal orders throughout history. Due to the secretive nature of such organizations, lack of supportive publicized material, and dubious motives for claiming Newton's participation in these groups, it is difficult to establish his actual membership in any specific organization.[25]

Regardless of his own membership status, Newton was a known associate of many individuals who themselves have often been labeled as members of various esoteric groups. It is unclear if these associations were a result of his being a well established and prominently publicized scholar, an early member and sitting President of The Royal Society (1703–1727), a prominent figure of State and Master of the Mint, a recognized Knight, or if Newton actually sought active membership within these esoteric organizations himself. Considering the nature and legality of alchemical practices during his lifetime, as well as his possession of various materials and manuscripts pertaining to alchemical research, Newton may very well have been a member of a group of like minded thinkers and colleagues. The organized level of this group (if in fact any existed), the level of their secrecy, as well as the depth of Newton's involvement within them, remains unclear.

Though Newton was largely considered a reclusive personality and not prone to socializing, during his lifetime being a member of "Societies" or "Clubs" was a very popular form of interpersonal networking. Considering his esteemed social status, it is probable that Newton would have had a least some contact with such groups at various levels. He was most certainly a member of The Royal Society of London for the Improvement of Natural Knowledge and the Spalding Gentlemen's Society,[26][27] however, these are considered learned societies, not esoteric societies. Newton's membership status within any particular secret society remains verifiably allusive and largely speculative, however, it still lends itself to popular sensationalism.

### Newton and The Rosicrucians

Perhaps the movement which most influenced Isaac Newton was Rosicrucianism.[28] Though the Rosicrucian movement had caused a great deal of excitement within Europe's scholarly community during the early seventeenth century, by the time Newton had reached maturity the movement had become less sensationalized. However, the Rosicrucian movement still would have a profound influence upon Newton, particularly in regard to his alchemical work and philosophical thought.

The Rosicrucian belief in being specially chosen for the ability to communicate with angels or spirits is echoed in Newton's prophetic beliefs. Additionally, the Rosicrucians proclaimed to have the ability to live forever through the use of the elixir vitae and the ability to produce limitless amounts of time and gold from the use of The Philosopher's Stone, which they claimed to have in their possession. Like Newton, the Rosicrucians were deeply religious, avowedly Christian, anti-Catholic, and highly politicised. <NAME> would have a deep interest in not just their alchemical pursuits, but also their belief in esoteric truths of the ancient past and the belief in enlightened individuals with the ability to gain insight into nature, the physical universe, and the spiritual realm.[28]

At the time of his death, <NAME> had 169 books on the topic of alchemy in his personal library, and was believed to have considerably more books on this topic during his Cambridge years, though he may have sold them before moving to London in 1696. For its time, his was considered one of the finest alchemical libraries in the world. In his library, Newton left behind a heavily annotated personal copy of The Fame and Confession of the Fraternity R.C., by <NAME> which represents an English translation of The Rosicrucian Manifestos. Newton also possessed copies of Themis Aurea and Symbola Aurea Mensae Duodecium by the learned alchemist <NAME>, both of which are significant early books about the Rosicrucian movement. These books were also extensively annotated by Newton.[28]

Newton's ownership of these materials by no means denotes membership within any early Rosicrucian order. Furthermore, considering that his personal alchemical investigations were focused upon discovering materials which the Rosicrucians professed to already be in possession of long before he was born, would seem to some to exclude Newton from their membership. However, in religious terms, the fact that a saint might have ‘found God’ would not preclude others from the search — quite the opposite. The Ancient & Mystical Order Rosae Crucis has always claimed Newton as a frater.[29] During his own life, Newton was openly 'accused' of being a Rosicrucian, as were many members of The Royal Society.[30] Though it is not known for sure if <NAME> was in fact a Rosicrucian, and he never publicly identified himself as one, from his writings it does appear that he may have shared many of their sentiments and beliefs.

## References

1. ^ <NAME>., "Newton, The Man"; Proceedings of the Royal Society Newton Tercentenary Celebrations, 15–19 July 1946; Cambridge University Press (1947)
2. ^ "Newton’s manuscripts provide evidence that he gave considerable thought to alchemy as emblematic of a purely scientific explanation of nature and was in fact deeply involved in conceiving alchemy as spiritual." <NAME>, "Some Modern Controversies on the Historiography of Alchemy" in Annual of Medieval Studies at CEU (2010), 186.
3. ^ "his alchemy cannot be seen solely in connection with his chemical experiments but was also a link between his religious beliefs and his scientific aims”. <NAME>, “Newton’s Alchemy,” in The Cambridge Companion to Newton, ed. I. <NAME> and <NAME> (Cambridge: Cambridge University Press, 2004), 375.
4. ^ a b c d Nova: Newton's Dark Secrets (2005). [1]. USA: PBS.
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10. ^ gallery
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14. ^ Goldish. Page 91.
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21. ^ Yahuda MS 7.3o, f. 8r
22. ^ Yahuda MS 7.3g, f. 13v
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• "The Foundations of Newton's Alchemy" by Sir <NAME> of the Royal Society[citation needed]