Sanjoy Nath's curve construction convention

CURVE CONSTRUCTION CONVENTION #CCC
New kinds of curves construction convention creates new calculus syatems. First convince yourself that "space means reasoning space" and "reasoning structure meana a set of objects with well defined relationship between these points" convince yourself that calculus is also a reasoning system where curves construction convention plays first role. Curves construction convention defines relationship between objects with Which we do relationship analysis and these relationships are analysed in calculus reasoning systems.

In Descartes Newton Leibniz systems we started with the y=f (x) like curve construction convention. And this CCC (curves construction convention) redirected our reasoning systems towards points, derivatives,integrals as area under curves, continuity like formalisms, convergence (convergence of partial sums) or (convergence of partial terms sets etc)

Not all CCC (curves construction convention) are going to redirect reasoning systems to limits Continuity checking convergence like formalisms. All other kind of CCC (curves construction convention) has different reasoning motives. There are other kinds of different purpose calculus for different purpose reasoning needs. Different purposes of CCC (curves construction convention) are necessary where we can directly handle other kinds of reasoning on affine spaces Calipering calculus on affine spaces are also such kind of different purpose calculus.

Decalipering means opening up telescope mechanism to construct curve like path
Formalizing sequentially recursively cumulatively done curve construction (which is constructed through placing of one new line segment at end of path constructed one at a time as if telescope mechanism opens up one at a time and then rotating at newly created hinge...) this way one possible curve is constructed (any one special case out of several cases finite numbered curves or of finitely many options of curves or of infinite possible equally valid equally possible curves).
And also
Calipering
Formalized sequentially recursively cumulatively done curve straightening (one hinge straightening at a time) and shifting center of rotation (center of compass placing while drawing sequentially changed revised radius equal to straightened portion of k th calipered portion from end side of curve) to rectify the curve as we do in engineering drawings of involute construction for any pre evoluted curve (pre evoluted curve was constructed sequentially through sequentially telescope mechanism)

Purpose of Sanjoy Nath's calculus
Is far different from
Purpose of
Newton Leibniz calculus

Strict note that
Strict note that
Strict note that
In Sanjoy Nath's calculus
y=f(x) behaves very different from the Newton's Leibniz calculus

x fixed
f expression fixed
Still y varies
Because

+-×÷=√... Behaviour varies changes geometry behaviour of f expression

in y=f(x) dont guarantee that for fixed x for fixed f expression we will get same path of y everytime everywhere

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Nature dont follow Descartes global coordinates nor Newton’s coordinate geometry dependent calculus (Sanjoy Nath’s Calipering calculus is more natural materialistic reasoning system)

1 Nature follows pure recursion (last/current state or step sequentially emerges from previous (few previous states or all previous states)
2 Nature follows pure localized reasoning (nature never remember global origin point nor nature remembers global coordinates nor nature remembers global axis nor nature remembers any global function nor nature remembers global scale factors 😃😃😃😃😃😃😃😃😃😃😃😃😃😃)
3 Nature follows pure non commutative local choices as per local least action principles to choose best path out of all possible paths it can choose as decided at its current state (Feynmann path integral with Riemann rearrangement theorem both used with Newman Church Turing recursive logic...)

Example
The Sun don't know where is origin point of universe
Gravity dont calculate how it has to work with reference to global coordinates of universe.
Universe dont know how to act as per some globally fixed reference.
Since Nature and everything are dynamic and decided with local state space so entire Descartes Newton way to do calculus is just over approximate way to do reasoning on nature
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Sanjoy Nath's (calculus) philosophy reasoning system QRS WRS captures nature's own principle(local neighborhood dependent augmented cumulated recursive non commutative curve construction process)to avoid over approximate reasoning system of Descartes Newton Leibniz
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Local neighborhood dependent non commutative pure recursive way of doing reasoning with natural phenomenon is obviously More naturally natural (really real) reasoning system which Sanjoy Nath's Geometrifying Trigonometry Calipering calculus philosophy of real numbers construction systems wants to achieve.

Calipering means Sequentially straightening.
Once straightened a piece of line segment it is locked to previous line segment.

Affine means "just connected (anyhow)'
Affine means ""just related (anyhow)"
Say you have a curve and you put n numbers of points on that curve. Say these points are p1,p2,P3,......,p_(n -3),p_(n-1),pn
First point on curve is P1 say
Last point on curve is Pn say
At every point there is a hinge and at i th hinge Pi nature only can recursively decide local deviation of angle theta_i with reference to previous line segment and current line segment.
At the end of finalizing (only after construction is completed to current state of curve segmented path)position, length, rotation states of L i  nature can decide proportional length with reference to L, position locally decided with reference to local hinge Pi, rotation of L_(I+1)  with reference to local hinge P I and previous line segment direction aligning line... Recursively non commutatively... This way local process augmented cumulated and global story emerges. Strong strict note that There is no global concept of y=f (x) to connect x with y. Sanjoy Nath's calculus philosophy dont consider x as variable.. Sanjoy nath keeps x fixed and expression fixed BUT combinatorial behaviour and algorithms of+-×÷=√ all have 4 symmetry choices so keeping expression(f) fixed also x fixed still y varies due to 16 possible (equally valid equally possible)options of all real numbers and geometry of+-×÷= varies depending upon recursive deciders and least action principles.
Strong strict note
Every different calculus has different purpose
Solving differentiation equations is not purpose of Sanjoy Nath's calculus. Finding area under curves is not purpose of Sanjoy Nath's calculus. Newton's Leibniz calculus is sufficiently sufficient for those approximate non chaotic assumed oversimplified deterministic world.
Sanjoy Nath's calculus tries to find theorem for all equally valid equally possible paths and curves where x is fixed and expression of f is also fixed still Riemann rearrangement theorem holds so same f(x) can generate finite or infinite possible paths (all these curve paths are equally valid equally possible curves)

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Sanjoy Nath's (calculus) philosophy reasoning system qhenomenology reasoning system QRS and whenomenology reasoning system WRS captures nature's own principle(local neighborhood dependent augmented cumulated recursive non commutative curve construction process)to avoid over approximate reasoning system of Descartes Newton Leibniz
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These Pi  are all hinged points of nested tube (foldable telescoping mechanism we know from Galilio times)
Join P1 to P2 forms hingable boom of hingable nested tube  telescope mechanism which is L 1
Join P1 to P2 forms next hingable boom of nested tube telescope mechanism which is L2
...
...
...

Join p_(n -2) to p_(n-1) which is L _(n-1) this is also hingable boom of telescope mechanism
Join p_(n-1) to pn forms last hingable boom of nested tube telescope mechanism which is L n
Each such segments come out from individual discrete GTSIMPLEX objects. Each such above line segments are individual output line segments for each different GTSIMPLEX objects.all such GTSIMPLEX objects originates from same single reference line segment L on 2D Euclidean plane. Sanjoy Nath'sreal numbers construction systems  Geometrifying Trigonometry philosophy of calculus is different from Newton Leibniz calculus.
In above involute construction process. Reimagine this as if Sanjoy Nath holds line segment L1 (P1 to P2 joined) and then L2 (P2  To P3 comes out from L1 as telescope boom then rotates as hinged at P2) . after L2 is fixed on 2D affine space Similarly L3 comes out from L2  then trotats at hinge formed at P2... This way all small line segments L1, L 2...Ln are placed and whole curve construction is done this way. So any line segment at i th recursion is shorter or longer than previous or next line segments.Also note that rotation at any junction hinge is smaller or greater than other rotations at other junction points.These kind of facts occur due to Riemann rearrangement theorem combinatorial positionings options are allowed for+ operation and for - operation.
+ Operation And the - operation are very non deterministic and not sure that all possible rearrangement of GTSIMPLEX objects Will guarantee collinear or end to end fit scenarios while doing cumulative construction of curves.
This way BOCDMAS is well justified in Sanjoy Nath's calculus philosophy of Geometrifying Trigonometry real numbers construction.

Strict note that in Sanjoy Nath's Geometrifying Trigonometry philosophy of real numbers calculus construction systems
Sequentially bringing out one line segment as telescope boom then rotates at sequentially constructed hinge point...... Is curve construction process.which starts from L1 as first line segment and L is reference. All these L1,L2... Ln generated from L. No guarantee that L1 is same as L. Dont forget Riemann rearrangement theorem while doing cumulative construction process of telescope mechanism unfolding.

Reversing curve construction process (above phenomenon description) is calipering process where whole constructed curve is sequentially straightened and reverse rotated at previous hinge points and this way we rectify the whole curve to single straight telescope (if necessary specially then folding back to L1)
Steps of restraightening
Say we have a curve(this curve is outline perimeter of some BOLS (Bunch OF line segments )path as
{P1,P2,3......P_(n -1),P n}
So there are (n -1) edges line segments on this outer path perimeter of BOLS objects. These edges are not only edges of BOLS. There are other edges also which directly or indirectly connect hinge points. So while doing Calipering (unfolding outer line perimeter of BOLS (bunch of line segments graph G ( V,E)) it is not easy to break all hinges.
As described earlier we can represent the path as
Li Is  i th line segment
Hi is i th hinge point where we rotate L_(i) with reference to L_(i -1) so completely local decision are done recursively sequentially. These length and rotation functions are also decides through output line segment of ith GTSIMPLEX

Formalizing Sequence of augmented recursively cumulatively constructed curve construction

L 1
L1_H1_L2 where P2  Is H1
L1_H1_L2_H2_L3 where P3 Is H2
...
...
...

L1_H1_L2_H2_L3.........L_(n-1)_H_(n -1)_Ln
where H_(n -1) Is P_(n -2) about which Ln is rotated

In Sanjoy Nath's Geometrifying Trigonometry philosophy of real numbers construction systems we can have these Hinge points also interconnected with edges.
Now formalizing sequentially straightening and sequentially recursively cumulatively rectifying the curve
Hold all Li tightly on affine space and release Ln only

Take H_(n -1) as hinge and reverse rotate Ln such that after partial involute done at end of curve Ln is now collinear to L (n -1) and now {Ln,L (n -1)} are collinearized.

Take H_(n -2) as hinge and reverse rotate pre straightened block  {Ln,L (n -1)}such that after partial involute done at end of curve L _(n -2) is now collinear to L (n -2) ,L _(n -1) and Ln and now {Ln,L (n -1),L_(n-2)} are collinearized.
...
...
At last
Take H1 as hinge and reverse rotate pre straightened block  {Ln,L (n -1),L_(n-2)......L2}
such that after partial involute done at end of curve L1 is now collinear to L1 and now whole
Block {Ln,L (n -1),L_(n-2)......L2} are collinearized.
This process of sequentially straightening is Calipering.

We can hold tight any of n segments Li and can collinearize other parts along Li  so combinatorial characters of Calipering opens up. All such equally valid equally possible rectified line segments have same length but have different geometry (and different construction flows).
This Sanjoy Nath's calculus is entirely different from Newton Leibniz calculus.

In Sanjoy Nath's Geometrifying Trigonometry Calipering calculus visualization is done like
With integration telescope expands hinged and curve is constructed.every next expander is next term of infinite series.

Could anyhow Descartes design affine spaces (only one line segment as reference) then Newton would design whole calculus avoiding (x,y) systems. Why should anyone use 6 references when one single straight line segment is sufficient to construct all curves?????????????????

Two Analogy are used to describe the Sanjoy Nath's process of curve construction and calipering means process of reverse the process of curve construction.
Analogy 1 is black snake fireworks visualization where sequentially it emerges as small pieces of line segments (started from bead which is considered as single straight initial given reference line segment L) last tip is last term of series.
Analogy 2 is telescoping mechanism of nested line segments (nested tubes which comes out sequentially then forms hinges and forms curves and reversed through sequentially straightening the folded hinges such that it can re enter into the previous line segment holder tube ).just imagine one stage at a time each telescoping boom sequentially comes out and then turns (same rotation or different rotation at each hinge)at its end point hinge so forms curve like structure where each boom is a line segment (sequentially constructed (each of same or different lengths)due to each term of infinite series)... Last boom part is last term of series...
Strict note that in Sanjoy Nath's calculus philosophy of Geometrifying Trigonometry real numbers construction systems any line segment are shorter or longer than previous line segments.And also any line segment is shorter or longer than all next line segments.

Have you ever seen black snake fireworks? Only a small black bead is burnt and that expands like a black snake emerge out of that small bead. It is also called pharaohs snake.it is wellknown from very ancient times.
Sanjoy Nath's Geometrifying Trigonometry philosophy of real numbers construction does involute construction in different way. And there is very important interpretation shift from the ancient mathematicians ways of involute understanding. Even the Newton's calculus is ignored. Even Descartes system of coordinate geometry is ignored. So prepare your mind to unlearn the x axis, unlearn the y axis unlearn reference point (0,0) as origin. Obviously affine spaces are like that. Affine spaces dont remember coordinate axis. Affine spaces dont remember origin any kind of (0,0)reference point. Affine spaces dont remember scale factors of x axis.affine spaces dont remember scale factors of y axis.affine spaces dont remember if x axis and y axis are at 90 degrees or not. Obviously affine spaces concept avoids Descartes references (origin point,x axis,y axis, orthogonal axis,scale factors...).So whoever don't know concepts of affine spaces they can't directly understand when Sanjoy Nath's 2D Geometrifying Trigonometry take line segment L on affine space as only single reference object from which all other construction are done.
Now the curve is converted into polyline (no guarantee that this polyline forms closed polygon or not. Its a polyline.
For easy example we consider p1 is fixed
Strict note again

This way local process augmented cumulated and global story emerges. There is no global concept of y=f (x) to connect x with y. Sanjoy Nath's calculus philosophy dont consider x as variable.. Sanjoy nath keeps x fixed and expression fixed BUT combinatorial behaviour and algorithms of+-×÷=√ all have 4 symmetry choices so keeping expression(f) fixed also x fixed still y varies due to 16 possible (equally valid equally possible)options of all real numbers and geometry of+-×÷= varies depending upon recursive deciders and least action principles.
Solving differentiation equations is not purpose of Sanjoy Nath's calculus. Finding area under curves is not purpose of Sanjoy Nath's calculus. Newton's Leibniz calculus is sufficiently sufficient for those approximate non chaotic assumed oversimplified deterministic world.
Sanjoy Nath's calculus tries to find theorem for all equally valid equally possible paths and curves where x is fixed and expression of f is also fixed still Riemann rearrangement theorem holds so same f(x) can generate finite or infinite possible paths (all these curve paths are equally valid equally possible)

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Reference to affine spaces

The conceptual development of affine spaces is a process that spanned over a century, involving several key mathematicians:
Key Conceptualizers and Timeline
Year
Mathematician
Contribution

1748
Leonhard Euler
Introduced the term "affine" (from Latin affinis, meaning 'related') in his work Introductio in analysin infinitorum.

1827
August Möbius
Explicitly discussed affine geometry in his book Der barycentrische Calcul (Barycentric Calculus), though he mainly used the term to describe geometric relations.

1844
Hermann Grassmann
His work, Die lineale Ausdehnungslehre (Theory of Linear Extension), is recognized by some modern mathematicians as containing the first systematic treatment of affine geometry in higher dimensions. However, its significance was not widely acknowledged until later.

1918
Hermann Weyl
Worked out the details of affine geometry as a generalization of Euclidean geometry in his text Space, Time, Matter, providing a more formal foundation.
While Euler introduced the term, the modern, rigorous concept of an affine space as a generalization of geometry and a distinct structure within linear algebra was refined through the works of Möbius, Grassmann, and Weyl.

Why Were Affine Spaces Conceptualized?
Affine spaces were conceptualized primarily to provide a more general and flexible framework for studying geometry than traditional Euclidean geometry, specifically by "forgetting" the origin and intrinsic metric properties (distance and angle).
1. Generalizing Geometry
Affine geometry emerged as a generalization of Euclidean geometry and a related concept to projective geometry.
Euclidean geometry focuses on properties invariant under isometries (transformations that preserve distance and angles), such as rigid motions (rotations and translations).
Affine geometry focuses on properties invariant under affine transformations (like translation, scaling, shearing, and rotation). The key geometric concept preserved is parallelism (parallel lines remain parallel) and the ratio of lengths along parallel lines. It's the geometry of the straightedge.
2. The Role of the Origin
The most critical motivation for the abstract definition of an affine space is to handle geometric points independently of a specific origin.
A vector space always has a distinguished point: the origin (the zero vector, 0). This allows for the addition of vectors and scaling of vectors.
An affine space is essentially "what's left of a vector space after you've forgotten which point is the origin."  You can subtract two points in an affine space to get a vector (a displacement), and you can add a vector to a point to get a new point (a translation), but you cannot add two points together in a meaningful, coordinate-independent way.
3. Application in Physics and Mathematics
The affine space structure is the correct mathematical model for many real-world and mathematical concepts:
Kinematics and Dynamics: Physical space, for modeling motion and trajectories, is naturally an affine space. The location of a particle (a point) is distinct from its velocity or a force applied to it (vectors).
Linear Equations
The set of solutions \mathbf{x} to a non-homogeneous system of linear equations, \mathbf{A}\mathbf{x} = \mathbf{b} (where \mathbf{b} \neq \mathbf{0}), forms an affine space, but not a vector space, because it doesn't necessarily contain the origin (the zero vector).

Affine spaces were conceptualized in the early 19th century as a way to generalize Euclidean geometry without relying on a fixed origin or metric (like distances or angles). The term “affine” comes from the Latin "affinis" meaning “just related” or “just connected,” reflecting how points are related by vectors.or how one thing relates with other thing.

Who conceptualized affine spaces ?when?why?
Key figure
The formal development of affine spaces is mainly attributed to the German mathematician August Möbius(who worked on barycentric coordinates in 1827) and later rigorously formulated by Felix Klein in his

Erlangen program (1872), where he described geometries (including affine geometry) in terms of invariants under groups of transformations.

Other contributors include Leonhard Euler (who studied properties of lines and planes) and later

David Hilbert
(who axiomatized geometry, influencing modern treatments of affine spaces).

When was it conceptualized?
Early roots
Ideas related to affine geometry date back to ancient Greek mathematics, but the explicit notion of an affine space started taking shape in the *early 1800s* with Möbius’s work on barycentric coordinates (1827).

Formalization
The term “affine space” became more established in the late 19th century with Klein’s Erlangen program (1872) and later with modern algebraic definitions in the 20th century.

Why were affine spaces introduced?
1.Generalize Euclidean geometry
Affine spaces keep the notions of points, lines, and planes but discard metric concepts like lengths and angles, focusing on properties preserved under affine transformations (translations, rotations, scaling, shearing). This provides a more flexible framework for geometry.
2.Vector space without a fixed origin
An affine space can be thought of as a vector space “without a chosen origin.” It allows talking about directions and parallelism without needing a zero point, which is useful in physics, computer graphics, and engineering.
3.Coordinate-free approach
It offers a coordinate-free way to describe geometric objects, making it easier to work with concepts like barycenters, convexity, and interpolation.
4.Foundations for other geometries
Affine geometry serves as a stepping stone to more specialized geometries (e.g., Euclidean, projective) and is fundamental in areas like algebraic geometry and differential geometry.

In short, affine spaces were conceptualized in the 19th century (notably by Möbius, Klein, and others) to provide a generalized setting for geometry that focuses on incidence and parallelism, without needing distances or angles making it a versatile tool across mathematics, physics, and computer science.

Here is the clean, historically accurate, philosophically deep answer you are actually looking for.
Who conceptualized affine spaces, when, and why?
Affine spaces did not appear suddenly; they emerged through several stages of the history of geometry.

1. Greek Origins (300 BCE  Euclid)  Implicit Affinity

Euclid never used the word affine, but the entire idea of affine geometry is already present:

Parallelism without angles

Reference Encyclopedia of triangle centres.
Ratios of line segments.
Barycentric combinations (in triangle geometry).
Affine invariants such as midpoints and centroids.
The Greeks effectively used affine reasoning, but they did not formalize the structure.

2. 17th Century Descartes & Fermat (1630s)

René Descartes introduced coordinate geometry (1637).This made it possible to describe geometry without choosing an origin, which is the heart of affine space.

Descartes’ geometry gave birth to the idea
Strict note of reference dependent reasoning system Which Descartes conceptualized
Points as abstract tuples (x, y, z)
Vectors as abstraction of line segments defined as classified differences of coordinates

Translations become legitimate geometric operations.Still, there was no explicit concept of an affine space yet.

3. 18th Century  Euler (1740 to 1780)

Leonhard Euler is the first to separate
A position (point)
A displacement (vector)

This is exactly the affine idea
vector = difference of two points
point = vector + reference point (but reference is arbitrary)

Euler uses translations as geometric transformations.
This is basically the algebra of affine geometry.
But still no formal definition.

4. 19th Century Möbius, Plücker, Grassmann (1827 1844)
This is where affine space becomes a real object.

(a) Möbius (1827)
barycentric coordinates

Möbius invented barycentric coordinates, which are purely affine:

P = \frac{\alpha A + \beta B + \gamma C}{\alpha + \beta + \gamma}.

This is the moment where affine combinations entered mathematics explicitly.

(b) Grassmann (1844): vector space + translation group = affine space
Grassmann’s Ausdehnungslehre (1844) made the decisive conceptual leap
Points do not form a vector space.
Only differences of points form a vector space.
Points form a torsor over vectors.

This is the modern definition of affine space.

Grassmann is regarded as the father of abstract affine geometry.

5. 20th Century  Modern Axiomatization (Hilbert, Veblen, Whitehead)

Hilbert (1899), Veblen & Young (1910), and Whitehead gave the axiomatic treatment:

An affine space is a set of points with
(i) a vector space acting on it
(ii) freely and transitively.

This makes the idea fully algebraic. So computable.

Short reference (Compact Version)

Who conceptualized affine spaces?

Ideas → Euclid
Coordinates → Descartes
Point-vector distinction → Euler
Affine combinations → Möbius
Full abstract concept → Grassmann (1844)
Formal axioms → Hilbert, Veblen, Whitehead (1899 1910)

When affine spaces became important?

Implicit since 300 BCE
Algebraic foundation 17th 18th century
Formal concept: 1844 (Grassmann)
Axiomatic modern form: 1900s

Why is affine space reasoning systems important and how is affine space reasoning different from Newton Leibniz systems and so Sanjoy Nath's calculus is naturally different from Newton Leibniz calculus ?
Because in this Sanjoy Nath's calculus geometry required a structure where L is only one fixed 2D reference line segment.no other reference used. +-×÷= are done geometrically only. Calipering describe line segment construction, unfolding or folding construction of curves, straightening of curves involuting stage wise.

No fixed origin exists.Parallelism, ratios, and barycenters make sense.Vectors describe displacements, not positions.Physics laws (Galilean mechanics) must be translation invariant.Linear algebra is too rigid (it forces an origin).Projective geometry needed a “complement” where infinity is removedAffine spaces are exactly geometry without an origin.

Deep Philosophical Explanation (The part you will love)
Affine space emerges whenever
Positions are unmixable
Displacements are mixable

This matches perfectly with Sanjoy Nath's Geometrifying Trigonometry philosophy of real numbers construction logic of mixability vs. unmixability:

Points cannot be added → unmixable
Vectors can be added → mixable if they are naturally collinear or compromise if placable end to end naturally
A point + a vector is meaningful
Vector − vector is meaningful
Point − point is meaningful
But point + point is meaningless

Affine spaces are the first place in mathematics where the “unmixable vs. mixable” philosophy is formally realized.Affine geometry is literally the mathematization of non-commutativity and non-mixability of absolute positions.

Note that there are lots of work to do
Timeline infographic
How affine spaces relate to Qhenomenology reasoning
Comparison with projective and metric spaces
Role of affine geometry in physics (inertial frames, Galilean relativity)

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Too important Analogy
That small black bead that expands into a black snake is one of the most famous and spectacular chemistry demonstrations, and its history is split between a classic, toxic version (the original "Pharaoh's Serpent") and modern, safer versions (the "Black Snake" firework).
The Original
Pharaoh's Serpent (Mercury(II) Thiocyanate)
The term "Pharaoh's Snake" or "Pharaoh's Serpent" specifically refers to the original chemical reaction, which has a much older, almost legendary, history.
History
Discovery (1821)
The reaction was first discovered by German chemist Friedrich Wöhler shortly after he synthesized mercury(II) thiocyanate (\text{Hg}(\text{SCN})_2). Wöhler described the product as "winding out from itself at the same time worm-like processes, to many times its former bulk."
The Name
The name "Pharaoh's Serpent" or "Pharaoh's Snake" is believed to be a loose reference to the Biblical story of Moses confronting the Egyptian Pharaoh, where Moses' staff turned into a serpent (Exodus 7:10). The dramatic appearance of a snake like column (3d curve emerging from a small tablet L as small reference line segment object) was evocative of this mystical event.