Apps/MaclaurinSpheroids/GoogleBooks - Revision history

Jet53man: /* Excerpts from A Treatise of Fluxions */

2021-09-13T15:48:28Z

Excerpts from A Treatise of Fluxions

← Older revision		Revision as of 11:48, 13 September 2021
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	{\| class="MaclaurinSpheroids" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"		{\| class="MaclaurinSpheroids" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
	\|-		\|-
	! style="height: 150px; width: 150px; background-color:#~~ffff99~~;" \|[[H_BookTiledMenu#Spheroidal_.26_Spheroidal-Like\|<b>Maclaurin's<br />Original Text<br />&<br />Analysis<br />(1742)</b>]]		! style="height: 150px; width: 150px; background-color:#ffeeee;" \|[[H_BookTiledMenu#Spheroidal_.26_Spheroidal-Like\|<b>Maclaurin's<br />Original Text<br />&<br />Analysis<br />(1742)</b>]]
	\|}		\|}
	Both Volume I and Volume II of Colin Maclaurin's "''A Treatise of Fluxions''" can now be accessed online via [https://books.google.com/books/about/A_Treatise_of_Fluxions.html?id=dCAOAAAAQAAJ Google Books]. (Check out [http://mathworld.wolfram.com/Fluxion.html Wolfram's explanation of the term, ''fluxion''.]) In what follows, we provide an abbreviated table of contents for both volumes and present selected excerpts from these volumes.		Both Volume I and Volume II of Colin Maclaurin's "''A Treatise of Fluxions''" can now be accessed online via [https://books.google.com/books/about/A_Treatise_of_Fluxions.html?id=dCAOAAAAQAAJ Google Books]. (Check out [http://mathworld.wolfram.com/Fluxion.html Wolfram's explanation of the term, ''fluxion''.]) In what follows, we provide an abbreviated table of contents for both volumes and present selected excerpts from these volumes.

Jet53man: /* See Also */

2021-09-13T00:54:13Z

Jet53man: /* Generate Curve AKG */

2021-09-13T00:40:53Z

Generate Curve AKG

← Older revision		Revision as of 20:40, 12 September 2021
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	In our above figure, this function, <math>~\mathrm{AQ}(z_\mathrm{RN})</math> normalized to <math>~a_1</math>, is delineated by the orange triangles over the coordinate range, <math>~-1 \le z_\mathrm{RN} \le 0</math>. If our interpretation of Maclaurin's discussion is correct, this is precisely the curve in Maclauin's Figure 291, No. 2 that is labeled, "AKG."		In our above figure, this function, <math>~\mathrm{AQ}(z_\mathrm{RN})</math> normalized to <math>~a_1</math>, is delineated by the orange triangles over the coordinate range, <math>~-1 \le z_\mathrm{RN} \le 0</math>. If our interpretation of Maclaurin's discussion is correct, this is precisely the curve in Maclauin's Figure 291, No. 2 that is labeled, "AKG."

	===Area Under the Curve===		===Area Under the Curve===

Jet53man: /* Prolate Spheroid */

2021-09-13T00:38:32Z

Prolate Spheroid

← Older revision		Revision as of 20:38, 12 September 2021
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	<table border="1" cellpadding="5" width="80%">		<table border="1" cellpadding="5" width="80%">
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	Figure extracted<sup>&dagger;</sup> from [https://books.google.com/books?id=xfQ7AQAAMAAJ&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false Colin Maclaurin (1742)]<~~p><~~/p>		Figure extracted<sup>&dagger;</sup> from [https://books.google.com/books?id=xfQ7AQAAMAAJ&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false Colin Maclaurin (1742)]<br />
	"''A Treatise of Fluxions''"<~~p><~~/p>		"''A Treatise of Fluxions''"<br />
	Volume II, Figure Plate XXXIII		Volume II, Figure Plate XXXIII
	</td></tr>		</td></tr>

Jet53man: /* Maclaurin's Discussion of Self-Gravitating, Oblate-Spheroidal Configurations */

2021-09-13T00:27:03Z

Maclaurin's Discussion of Self-Gravitating, Oblate-Spheroidal Configurations

← Older revision		Revision as of 20:27, 12 September 2021
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	<table border="1" cellpadding="5" width="80%">		<table border="1" cellpadding="5" width="80%">
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	Paragraph (and related figure) extracted<sup>&dagger;</sup> from [https://books.google.com/books?id=xfQ7AQAAMAAJ&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false Colin Maclaurin (1742)]<~~p><~~/p>		Paragraph (and related figure) extracted<sup>&dagger;</sup> from [https://books.google.com/books?id=xfQ7AQAAMAAJ&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false Colin Maclaurin (1742)]<br />
	"''A Treatise of Fluxions''"<~~p><~~/p>		"''A Treatise of Fluxions''"<br />
	Volume II, Chapter XIV, §628		Volume II, Chapter XIV, §628
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	[[File:frustums.jpg\|link=http://www.mathalino.com/reviewer/derivation-of-formulas/derivation-of-formula-for-volume-of-a-frustum\|300px\|center\|Diagram ~~reposted~~ from MATHalino.com]]		[[File:frustums.jpg\|link=http://www.mathalino.com/reviewer/derivation-of-formulas/derivation-of-formula-for-volume-of-a-frustum\|300px\|center\|To be inserted: Diagram from MATHalino.com]]
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	==Interpreting Maclaurin's Key Concluding Theorem==		==Interpreting Maclaurin's Key Concluding Theorem==

Jet53man: /* Excerpts from A Treatise of Fluxions */

2021-09-13T00:12:30Z

Excerpts from A Treatise of Fluxions

← Older revision		Revision as of 20:12, 12 September 2021
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	! style="height: 150px; width: 150px; background-color:#ffff99;" \|[[H_BookTiledMenu#Spheroidal_.26_Spheroidal-Like\|<b>Maclaurin's<br />Original Text<br />&<br />Analysis<br />(1742)</b>]]		! style="height: 150px; width: 150px; background-color:#ffff99;" \|[[H_BookTiledMenu#Spheroidal_.26_Spheroidal-Like\|<b>Maclaurin's<br />Original Text<br />&<br />Analysis<br />(1742)</b>]]
	\|}		\|}

	Both Volume I and Volume II of Colin Maclaurin's "''A Treatise of Fluxions''" can now be accessed online via [https://books.google.com/books/about/A_Treatise_of_Fluxions.html?id=dCAOAAAAQAAJ Google Books]. (Check out [http://mathworld.wolfram.com/Fluxion.html Wolfram's explanation of the term, ''fluxion''.]) In what follows, we provide an abbreviated table of contents for both volumes and present selected excerpts from these volumes.		Both Volume I and Volume II of Colin Maclaurin's "''A Treatise of Fluxions''" can now be accessed online via [https://books.google.com/books/about/A_Treatise_of_Fluxions.html?id=dCAOAAAAQAAJ Google Books]. (Check out [http://mathworld.wolfram.com/Fluxion.html Wolfram's explanation of the term, ''fluxion''.]) In what follows, we provide an abbreviated table of contents for both volumes and present selected excerpts from these volumes.

	<table border="1" align="center">		<table border="1" align="center" width="50%">
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	<th align="center" colspan="2">		<th align="center" colspan="2">
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	<td align="center>		<td align="center" width="50%">
	[[File:TitlePage1stEdition01.png\|center\|border\|200px\|Volume I (1742)]]		[[File:TitlePage1stEdition01.png\|center\|border\|200px\|Volume I (1742)]]
	</td>		</td>
	<td align="center>		<td align="center">
	[[File:TitlePage2ndEdition01.png\|center\|border\|200px\|Volume II (2<sup>nd</sup> Ed., 1801)]]		[[File:TitlePage2ndEdition01.png\|center\|border\|200px\|Volume II (2<sup>nd</sup> Ed., 1801)]]
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	As is illustrated by the images displayed in the following two panels (click in one of the panels in order to view a higher-resolution image), each published "Figure Plate" was digitized (by Google Books) in segments that must be digitally pieced together in order to reconstruct some of the individual figures. See, for example, the discussion that references [[#Paragraph630\|Fig. 282, below]].		As is illustrated by the images displayed in the following two panels (click in one of the panels in order to view a higher-resolution image), each published "Figure Plate" was digitized (by Google Books) in segments that must be digitally pieced together in order to reconstruct some of the individual figures. See, for example, the discussion that references [[#Paragraph630\|Fig. 282, below]].


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	[[File:Vol2Paragraph641.png\|400px\|thumb\|center\|Extracted directly from §641 of Maclaurin's Book 1, as digitized by Google]]		[[File:Vol2Paragraph641.png\|400px\|thumb\|center\|Extracted directly from §641 of Maclaurin's Book 1, as digitized by Google]]

	Here we assess this ''geometrically'' formulated statement using today's more common differential operators and algebraic expressions. In particular, we draw from the [[~~User:Tohline/~~Apps/MaclaurinSpheroids#Maclaurin_Spheroids_.28axisymmetric_structure.29\|discussion of Maclaurin spheroids that has been presented in an accompanying chapter]] of this H_Book.		Here we assess this ''geometrically'' formulated statement using today's more common differential operators and algebraic expressions. In particular, we draw from the [[Apps/MaclaurinSpheroids#Maclaurin_Spheroids_.28axisymmetric_structure.29\|discussion of Maclaurin spheroids that has been presented in an accompanying chapter]] of this H_Book.

	<font color="maroon">'''CA'''</font>: This is the semi-minor (polar) axis of the spheroid, which we have called, <math>~a_3</math>.		<font color="maroon">'''CA'''</font>: This is the semi-minor (polar) axis of the spheroid, which we have called, <math>~a_3</math>.
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	This is precisely the equilibrium frequency that we have [[~~User:Tohline/~~Apps/MaclaurinSpheroids#EquilibriumFrequency\|derived elsewhere]], from an evaluation of the gravitational potential of uniform-density spheroids.		This is precisely the equilibrium frequency that we have [[Apps/MaclaurinSpheroids#EquilibriumFrequency\|derived elsewhere]], from an evaluation of the gravitational potential of uniform-density spheroids.

	<!-- ALSO COULD WRITE THE FOLLOWING ...		<!-- ALSO COULD WRITE THE FOLLOWING ...
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	and, because it is [[~~User:Tohline/~~Appendix/Ramblings/ToroidalCoordinates#Off-center_Circle\|off-center]], the green quarter circle is defined by the relation,		and, because it is [[Appendix/Ramblings/ToroidalCoordinates#Off-center_Circle\|off-center]], the green quarter circle is defined by the relation,
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	[[File:Vol2Par644KeyPhrase.png\|400px\|thumb\|center\|Extracted directly from §644 of Maclaurin's Book 1, as digitized by Google]]		[[File:Vol2Par644KeyPhrase.png\|400px\|thumb\|center\|Extracted directly from §644 of Maclaurin's Book 1, as digitized by Google]]

	We interpret this phrase to mean that, as the angle, <math>~\alpha</math>, varies from <math>~\tfrac{\pi}{2}</math> to <math>~0</math>, the curve "AKG" is generated by plotting, as the ordinate, the length of the line-segment "AQ" and, as the abscissa, the <math>~z</math>-coordinate location of the vertical line, "RN" — we'll label this, <math>~z_\mathrm{RN}</math>. In our terminology, as [[~~User:Tohline/~~Apps/MaclaurinSpheroids/GoogleBooks#Intersection_With_Circle\|derived above]], <math>~z_\mathrm{RN}</math> is the <math>~z</math>-coordinate of the intersection of the red line-segment with the green quarter-circle, that is,		We interpret this phrase to mean that, as the angle, <math>~\alpha</math>, varies from <math>~\tfrac{\pi}{2}</math> to <math>~0</math>, the curve "AKG" is generated by plotting, as the ordinate, the length of the line-segment "AQ" and, as the abscissa, the <math>~z</math>-coordinate location of the vertical line, "RN" — we'll label this, <math>~z_\mathrm{RN}</math>. In our terminology, as [[Apps/MaclaurinSpheroids/GoogleBooks#Intersection_With_Circle\|derived above]], <math>~z_\mathrm{RN}</math> is the <math>~z</math>-coordinate of the intersection of the red line-segment with the green quarter-circle, that is,
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	And the magnitude of "AQ" is obtained from the <math>~z</math>-coordinate location of the vertical line, "MQ" — that is, the <math>~z</math>-coordinate of the intersection of the red line-segment with the blue ellipse — as [[~~User:Tohline/~~Apps/MaclaurinSpheroids/GoogleBooks#Intersection_With_Ellipse\|derived above]]. Specifically, we interpret Maclaurin's phrase to mean that,		And the magnitude of "AQ" is obtained from the <math>~z</math>-coordinate location of the vertical line, "MQ" — that is, the <math>~z</math>-coordinate of the intersection of the red line-segment with the blue ellipse — as [[Apps/MaclaurinSpheroids/GoogleBooks#Intersection_With_Ellipse\|derived above]]. Specifically, we interpret Maclaurin's phrase to mean that,
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	This is precisely the expression for the coefficient, <math>~A_1</math>, that is [[~~User:Tohline/~~ThreeDimensionalConfigurations/HomogeneousEllipsoids#Prolate_Spheroids_.28a1_.3E_a2_.3D_a3.29\|traditionally used in the expression for the gravitational potential inside homogeneous, prolate spheroids]].		This is precisely the expression for the coefficient, <math>~A_1</math>, that is [[ThreeDimensionalConfigurations/HomogeneousEllipsoids#Prolate_Spheroids_.28a1_.3E_a2_.3D_a3.29\|traditionally used in the expression for the gravitational potential inside homogeneous, prolate spheroids]].


			{{SGFworkInProgress}}

	~~{{LSU_WorkInProgress}}~~
	===Comparison With More Traditional Derivation===		===Comparison With More Traditional Derivation===

	In our [[~~User:Tohline/~~ThreeDimensionalConfigurations/HomogeneousEllipsoids#Acceleration_at_the_Pole\|separate derivation of the gravitational acceleration at the pole of a prolate spheroid]] that uses a more traditional approach, we obtained this key coefficient by integrating over a different function, namely,		In our [[ThreeDimensionalConfigurations/HomogeneousEllipsoids#Acceleration_at_the_Pole\|separate derivation of the gravitational acceleration at the pole of a prolate spheroid]] that uses a more traditional approach, we obtained this key coefficient by integrating over a different function, namely,
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	We should be able to demonstrate explicitly that these two integrands are the same. We begin by recognizing that the relationship between the dimensionless coordinate, <math>~z/a_1</math>, that we have used to decipher Maclaurin's presentation and the dimensionless coordinate, <math>~\zeta</math>, that was [[~~User:Tohline/~~ThreeDimensionalConfigurations/HomogeneousEllipsoids#Acceleration_at_the_Pole\|used in our separate analysis]], is,		We should be able to demonstrate explicitly that these two integrands are the same. We begin by recognizing that the relationship between the dimensionless coordinate, <math>~z/a_1</math>, that we have used to decipher Maclaurin's presentation and the dimensionless coordinate, <math>~\zeta</math>, that was [[ThreeDimensionalConfigurations/HomogeneousEllipsoids#Acceleration_at_the_Pole\|used in our separate analysis]], is,
	<div align="center">		<div align="center">
	<math>\frac{z}{a_1} = -\zeta \, .</math>		<math>\frac{z}{a_1} = -\zeta \, .</math>
	</div>		</div>

	Returning to the first expression in the above subsection that details the [[~~User:Tohline/~~Apps/MaclaurinSpheroids/GoogleBooks#Intersection_With_Ellipse\|intersection of the red line-segment with the blue ellipse]], we can therefore write,		Returning to the first expression in the above subsection that details the [[Apps/MaclaurinSpheroids/GoogleBooks#Intersection_With_Ellipse\|intersection of the red line-segment with the blue ellipse]], we can therefore write,
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	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Jet53man: Created page with "FORCETOC =Excerpts from A Treatise of Fluxions= {| class="MaclaurinSpheroids" style="float:left; margin-right: 20px; border-style: s..."

2021-09-12T23:59:04Z

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	=See Also=		=See Also=
	* Our review of the [[~~User:Tohline/~~ThreeDimensionalConfigurations/HomogeneousEllipsoids#Properties_of_Homogeneous_Ellipsoids\|Properties of Homogeneous Ellipsoids]]		* Our review of the [[ThreeDimensionalConfigurations/HomogeneousEllipsoids#Properties_of_Homogeneous_Ellipsoids\|Properties of Homogeneous Ellipsoids]]
	* Our derivation of the properties of [[~~User:Tohline/~~Apps/MaclaurinSpheroids#Maclaurin_Spheroids_.28axisymmetric_structure.29\|Maclaurin Spheriods]]		* Our derivation of the properties of [[Apps/MaclaurinSpheroids#Maclaurin_Spheroids_.28axisymmetric_structure.29\|Maclaurin Spheriods]]
	* Wikipedia's documentation of [https://en.wikipedia.org/wiki/Colin_Maclaurin Colin Maclaurin]		* Wikipedia's documentation of [https://en.wikipedia.org/wiki/Colin_Maclaurin Colin Maclaurin]
	* Chandrasekhar's (1967) [http://people.ucsc.edu/~igarrick/EART290/chandrasekhar_1967.pdf historical account]		* Chandrasekhar's (1967) [http://people.ucsc.edu/~igarrick/EART290/chandrasekhar_1967.pdf historical account]
	* [http://www.jstor.org/stable/3603342?seq=1#page_scan_tab_contents Notes on the Life and Works of colin Maclaurin] by Charles Tweedie (1919)		* [http://www.jstor.org/stable/3603342?seq=1#page_scan_tab_contents Notes on the Life and Works of colin Maclaurin] by Charles Tweedie (1919)
	* <span id="Todhunter1873">[http://ebooks.library.cornell.edu/cgi/t/text/pageviewer-idx?c=math;cc=math;idno=05710001;view=image;seq=5 I. Todhunter (1873)], ''A History of the Mathematical Theories of Attraction and the Figure of the Earth, from the time of Newton to that of Laplace.''</span> Note that Todhunter's discussion of Maclaurin's contributions can be found in Chapter IX, pp. 133-175.		* <span id="Todhunter1873">[http://ebooks.library.cornell.edu/cgi/t/text/pageviewer-idx?c=math;cc=math;idno=05710001;view=image;seq=5 I. Todhunter (1873)], ''A History of the Mathematical Theories of Attraction and the Figure of the Earth, from the time of Newton to that of Laplace.''</span> Note that Todhunter's discussion of Maclaurin's contributions can be found in Chapter IX, pp. 133-175.
			<!--
	[[File:Todhunter1873Sect252.png\|400px\|thumb\|center\|Extracted directly from §252 of Todhunter's book, as digitized by xxx]]		[[File:Todhunter1873Sect252.png\|400px\|thumb\|center\|Extracted directly from §252 of Todhunter's book, as digitized by xxx]]
			-->

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			<div align="center">Extracted from §252 of [http://ebooks.library.cornell.edu/cgi/t/text/pageviewer-idx?c=math;cc=math;idno=05710001;view=image;seq=5 I. Todhunter (1873)]</div>
			<font color="darkgreen">
			<b>252.</b>   "Maclaurin then in his Articles 644 … 647 investigates accurate expressions for the attraction of any ellipsoid of revolution on a particle at the pole or at the equator. The investigations are conducted in the manner of the time by representing the attractions by the areas of certain curves, and finding the areas by the method of fluents. The results agree with those obtained by analysis, and presented in modern works on Statics. Maclaurin's processes are remarkable specimens of ingenuity, considering the date of their publication; but they will not be very interesting to a modern reader."
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Apps/MaclaurinSpheroids/GoogleBooks - Revision history

Jet53man: /* Excerpts from A Treatise of Fluxions */

Jet53man: /* See Also */

Jet53man: /* Generate Curve AKG */

Jet53man: /* Prolate Spheroid */

Jet53man: /* Maclaurin's Discussion of Self-Gravitating, Oblate-Spheroidal Configurations */

Jet53man: /* Excerpts from A Treatise of Fluxions */

Jet53man: Created page with "__FORCETOC__ =Excerpts from A Treatise of Fluxions= {| class="MaclaurinSpheroids" style="float:left; margin-right: 20px; border-style: s..."

Jet53man: Created page with "FORCETOC =Excerpts from A Treatise of Fluxions= {| class="MaclaurinSpheroids" style="float:left; margin-right: 20px; border-style: s..."