Making sense of William Jones's solution of quadratic equations and notationWilliam Jones, when discussing quadratic equations, says: "Therefore if ∨ be put for the Sign of any Term, and ∧ for the contrary, all Forms of Quadratics with their Solutions, will be reduc'd to this one. If xx∨ax∨b=0 then ∧12a±aa∧b―!,∣12."Could you please help me make sense:(a) What is meant with "the contrary" of "any sign"?(b) Do I understand correctly that the "then" part supposed to be a solution to the preceding quadratic equation? If yes, how does this work exactly? How does it align with how we would symbolize the solution today?

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Making sense of William Jones&#039;s solution of quadratic equations and notationWilliam Jones, when discussing quadratic equations, says: &quot;Therefore if ∨  be put for the Sign of any Term, and ∧ for the contrary, all Forms of Quadratics with their Solutions, will be reduc&#039;d to this one. If xx∨ax∨b=0 then ∧12a±aa∧b―!,∣12.&quot;Could you please help me make sense:(a) What is meant with &quot;the contrary&quot; of &quot;any sign&quot;?(b) Do I understand correctly that the &quot;then&quot; part supposed to be a solution to the preceding quadratic equation? If yes, how does this work exactly? How does it align with how we would symbolize the solution today?

Raiden Griffin · Accepted Answer

By &quot;contrary&quot;, it looks like he just means &quot;opposite sign&quot; - so if ∨ is &quot;+&quot; then land is &quot;-&quot; and vice versa.In other words, this is saying that the solutions to x2+ax+b=0 are −12a+a2−b and similarly the solutions to x2−ax−b=0 are +12a+a2+b, although it looks like there&#039;s a factor missing somewhere that would actually make the expression correct (there might also be some grouping involved in the way the notation works that I haven&#039;t picked up on).

Cassius Villarreal · Accepted Answer

The &quot;∨&quot; and &quot;∧&quot; appear to play the role of modern &quot;±&quot; and &quot;∓&quot;, assigned independently to a and b. The &quot;contrary&quot; of a sign is simply its opposite.So, I would translate the example equation like this:×∨ax∨b=0&amp;nbsp;→&amp;nbsp;x2±1ax±2b=0(I&#039;ve placed subscripts on the &quot;±&quot;s to indicate that they&#039;re independent signs.)By the Quadratic Formula, we &quot;know&quot; the solution isx&amp;nbsp;=&amp;nbsp;12−(±1a)±(±1a)2−4(±2b)&amp;nbsp;=&amp;nbsp;12∓1a±a2∓24b&amp;nbsp;(1)Translating this back to the other form is a little tricky (there may be a typo, or I may be misinterpreting the notation), but even here we notice that the original &quot;±&quot;s have become &quot;∓&quot;s (except for the squared sign that just disappears); thus, &quot;∨&quot;s have become &quot;∧&quot;s. Expanding (1), we getx&amp;nbsp;=&amp;nbsp;12−(±1a)±(±1a)2−4(±2b)&amp;nbsp;=&amp;nbsp;12∓1a±a2∓24b&amp;nbsp;(1)This almost matches the notation∧12a;±;(aa∧b)12&amp;nbsp;(3)&amp;nbsp;except the&amp;nbsp;14&amp;nbsp;seems to have gone missing from the&amp;nbsp;a2&amp;nbsp;(or&amp;nbsp;12&amp;nbsp;is missing from the a that&#039;s being squared).

Making sense of William Jones's solution of quadratic

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