# Secant line
A secant line is a straight line that cuts a curve at two (or more) points; on the [graph of a function](https://www.edgepedia.org/graph-of-a-function) \( y = f(x) \), it is the line through two points \( (a, f(a)) \) and \( (b, f(b)) \) with \( a \neq b \)[1](https://ximera.osu.edu/precal/PrecWRev2Unit11/11-4SecantLinesRevisited/11-4-1AverageRateofChangeSecantLines) • [2](https://mathworld.wolfram.com/SecantLine.html). The word comes from the Latin *secare*, to cut: a secant crosses the curve, while a tangent touches it at a single point. Secant lines matter for two reasons: their slope measures the average rate of change of a function over an interval, and their limiting position as the two points merge is the tangent line, whose slope is the derivative[1](https://ximera.osu.edu/precal/PrecWRev2Unit11/11-4SecantLinesRevisited/11-4-1AverageRateofChangeSecantLines) • [3](https://openstax.org/books/calculus-volume-1/pages/3-1-defining-the-derivative).
| Key fact | Detail |
|---|---|
| Definition | A line through two distinct points \( (a, f(a)) \) and \( (b, f(b)) \) on the graph of \( f \)[1](https://ximera.osu.edu/precal/PrecWRev2Unit11/11-4SecantLinesRevisited/11-4-1AverageRateofChangeSecantLines) |
| Slope | \( m_{\mathrm{sec}} = \dfrac{f(b) - f(a)}{b - a} \), the average rate of change of \( f \) on \( [a, b] \)[1](https://ximera.osu.edu/precal/PrecWRev2Unit11/11-4SecantLinesRevisited/11-4-1AverageRateofChangeSecantLines) • [2](https://mathworld.wolfram.com/SecantLine.html) |
| Limit | As the second point approaches the first, the secant lines approach the tangent line, and the limit of their slopes is the derivative \( f'(a) \)[3](https://openstax.org/books/calculus-volume-1/pages/3-1-defining-the-derivative) |
| Difference quotient | \( \dfrac{f(a+h) - f(a)}{h} \) is the slope of the secant through \( (a, f(a)) \) and \( (a+h, f(a+h)) \)[3](https://openstax.org/books/calculus-volume-1/pages/3-1-defining-the-derivative) |
| Numerical use | The secant method replaces Newton's derivative with a secant slope over the two most recent iterates, needing two starting values but no derivative[4](https://persson.berkeley.edu/math128a/lecture_notes/math128a_lecture_notes_chapter_2.html) |
| Convergence order | The secant method converges with order \( \varphi = (1 + \sqrt{5})/2 \approx 1.618 \), between linear and quadratic[5](https://www.cl.cam.ac.uk/teaching/0506/NumAnII/na2seq2.pdf) |
| Classical geometry | In a circle, two secants through an interior point satisfy Euclid III.35: the products of the segments are equal[6](https://www.euclids-elements.org/elements/books/bookIII/propositions/propIII35/) |
## Slope and average rate of change
For a function \( f \) defined on an interval \( [a, b] \), the average rate of change is
\[ \frac{f(b) - f(a)}{b - a}, \]
and this number is exactly the slope of the secant line joining \( (a, f(a)) \) to \( (b, f(b)) \)[1](https://ximera.osu.edu/precal/PrecWRev2Unit11/11-4SecantLinesRevisited/11-4-1AverageRateofChangeSecantLines). If \( f \) measures distance against time, the secant slope is the average velocity over the interval; if \( f \) measures cost against output, it is the average cost per unit over that range.
Writing the second point as \( a + h \) gives the same slope in the form of the difference quotient,
\[ m_{\mathrm{sec}} = \frac{f(a + h) - f(a)}{h}, \]
which is the standard starting point for defining the derivative[3](https://openstax.org/books/calculus-volume-1/pages/3-1-defining-the-derivative). A worked example shows the mechanics: for \( f(x) = x^2 \) with points at \( x = 1 \) and \( x = 2 \), the secant through \( (1,1) \) and \( (2,4) \) has slope \( (4-1)/(2-1) = 3 \) and equation \( y = 3x - 2 \)[1](https://ximera.osu.edu/precal/PrecWRev2Unit11/11-4SecantLinesRevisited/11-4-1AverageRateofChangeSecantLines).
## From secant to tangent
As \( h \) shrinks toward 0, the second point \( (a+h, f(a+h)) \) slides toward \( (a, f(a)) \), and the connecting line approaches the tangent to the curve at \( x = a \); its slope approaches the slope of that tangent[7](https://openstax.org/books/precalculus-2e/pages/12-4-derivatives). Formally, the tangent line at \( a \) is the line through \( (a, f(a)) \) with slope
\[ m_{\mathrm{tan}} = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}, \]
provided the limit exists[3](https://openstax.org/books/calculus-volume-1/pages/3-1-defining-the-derivative). This limit is the derivative \( f'(a) \), the instantaneous rate of change of \( f \) at \( a \)[3](https://openstax.org/books/calculus-volume-1/pages/3-1-defining-the-derivative). In this sense the derivative is defined as the limit of secant slopes, and for a sufficiently smooth one-variable function the limiting position of the secant lines to its graph is the tangent line[8](https://arxiv.org/abs/2208.13639).
The same idea extends to several variables. For a function \( f(x, y) \), the slope of the secant through \( (x, y) \) and \( (x+h, y) \) represents the average rate of change of \( f \) as one travels parallel to the \( x \)-axis, and its limit is the partial derivative with respect to \( x \)[9](https://openstax.org/books/calculus-volume-3/pages/4-3-partial-derivatives).
## Secants in classical geometry
In [Euclidean geometry](https://www.edgepedia.org/euclidean-geometry) a secant to a circle is a line that meets the circle at two points. Euclid's *Elements*, Book III, Proposition 35 states that if two chords (segments of secants) cut one another inside a circle, the rectangle contained by the segments of one equals the rectangle contained by the segments of the other: for chords \( AC \) and \( BD \) meeting at \( E \), \( AE \cdot EC = DE \cdot EB \)[6](https://www.euclids-elements.org/elements/books/bookIII/propositions/propIII35/). The companion result for a secant and a tangent drawn from an external point appears as the final proposition of Book III: the square on the tangent equals the rectangle on the whole secant and its external segment[10](https://www.hlevkin.com/hlevkin/90MathPhysBioBooks/mathHistory/Boyer-AHistoryOfMathematics.pdf).
## The secant method
Secant lines also power a widely used root-finding algorithm. [Newton's method](https://www.edgepedia.org/newtons-method) iterates \( x_{k+1} = x_k - f(x_k)/f'(x_k) \), which requires the derivative \( f' \). The secant method replaces that derivative with the slope of the secant through the two most recent iterates,
\[ f'(p_{n-1}) \approx \frac{f(p_{n-1}) - f(p_{n-2})}{p_{n-1} - p_{n-2}}, \]
giving
\[ p_n = p_{n-1} - \frac{f(p_{n-1})(p_{n-1} - p_{n-2})}{f(p_{n-1}) - f(p_{n-2})}. \]
Geometrically, each new iterate is where the line through the two previous points meets the axis: Newton's tangent is replaced by a secant[4](https://persson.berkeley.edu/math128a/lecture_notes/math128a_lecture_notes_chapter_2.html). The method needs two starting values but no derivative[4](https://persson.berkeley.edu/math128a/lecture_notes/math128a_lecture_notes_chapter_2.html).
**Convergence order.** Near a simple root \( \alpha \), with errors \( e_n = x_n - \alpha \), a Taylor expansion shows that the errors satisfy
\[ |e_{n+1}| \approx M \, |e_n| \, |e_{n-1}|, \qquad M = \left| \frac{f''(\alpha)}{2 f'(\alpha)} \right|, \]
the same constant \( M \) as in Newton's method[5](https://www.cl.cam.ac.uk/teaching/0506/NumAnII/na2seq2.pdf). Seeking an order \( p \) with \( |e_{n+1}| \sim C|e_n|^p \) forces \( p^2 - p - 1 = 0 \), whose positive solution is the golden ratio \( p = (1 + \sqrt{5})/2 \approx 1.618 \)[5](https://www.cl.cam.ac.uk/teaching/0506/NumAnII/na2seq2.pdf). This is superlinear convergence, between the linear order of simpler methods and the quadratic order \( p = 2 \) of Newton's method[11](https://www.unioviedo.es/compnum/labs/lab04_ne/lab04_ne.html).
**Efficiency trade-off.** Newton's method converges faster per iteration but costs two function evaluations per step (the function and its derivative), while the secant method reuses the previous evaluation and costs one[4](https://persson.berkeley.edu/math128a/lecture_notes/math128a_lecture_notes_chapter_2.html) • [12](https://akbis.gaziantep.edu.tr/yonetim/upload/files/38768-6012.pdf). Comparing efficiency as convergence order per evaluation gives \( 2^{1/2} \approx 1.41 \) for Newton and \( \varphi^{1/1} \approx 1.62 \) for the secant method, so the secant method is often the better bargain in practice despite taking more steps for the same accuracy[4](https://persson.berkeley.edu/math128a/lecture_notes/math128a_lecture_notes_chapter_2.html) • [13](https://www.theorempath.com/topics/secant-method).
**Limitations.** The convergence argument is local: it holds only for starting guesses near the root, and the method needs two initial values[14](https://andreask.cs.illinois.edu/cs450-s26/notes.pdf). It also converges faster than bisection or regula falsi, though Newton can outperform it when derivatives are cheap[15](https://web.sha1.bfh.science/NumMethods/NumMethods.pdf). The related method of false position (regula falsi) works like the secant method but adds a test to keep the root bracketed between iterates[4](https://persson.berkeley.edu/math128a/lecture_notes/math128a_lecture_notes_chapter_2.html).
## Related uses
The same two-point construction appears in numerical search over sorted data: secant search estimates the next position to check from the values at the two previous positions, using the secant slope to approximate the derivative in the Newton-Raphson formula, and falls back to a midpoint step when the estimate leaves the search range[16](https://xlinux.nist.gov/dads/HTML/secantSearch.html). More broadly, secant-based derivative-free updates underlie quasi-Newton methods in optimization, a family to which the secant method itself belongs[14](https://andreask.cs.illinois.edu/cs450-s26/notes.pdf).
## References
1. [Average Rate of Change and Secant Lines, Ximera (Ohio State University)](https://ximera.osu.edu/precal/PrecWRev2Unit11/11-4SecantLinesRevisited/11-4-1AverageRateofChangeSecantLines)
2. [Secant Line, Wolfram MathWorld](https://mathworld.wolfram.com/SecantLine.html)
3. [3.1 Defining the Derivative, Calculus Volume 1, OpenStax](https://openstax.org/books/calculus-volume-1/pages/3-1-defining-the-derivative)
4. [Chapter 2, Solutions of Equations in One Variable, Math 128A lecture notes, UC Berkeley](https://persson.berkeley.edu/math128a/lecture_notes/math128a_lecture_notes_chapter_2.html)
5. [Numerical Analysis II, University of Cambridge (rate of convergence of Newton and secant methods)](https://www.cl.cam.ac.uk/teaching/0506/NumAnII/na2seq2.pdf)
6. [Euclid's Elements, Book III, Proposition 35](https://www.euclids-elements.org/elements/books/bookIII/propositions/propIII35/)
7. [12.4 Derivatives, Precalculus 2e, OpenStax](https://openstax.org/books/precalculus-2e/pages/12-4-derivatives)
8. [The surface tangent paradox and the difference vector quotient of a secant plane, arXiv](https://arxiv.org/abs/2208.13639)
9. [4.3 Partial Derivatives, Calculus Volume 3, OpenStax](https://openstax.org/books/calculus-volume-3/pages/4-3-partial-derivatives)
10. [A History of Mathematics (Boyer), section on Euclid of Alexandria](https://www.hlevkin.com/hlevkin/90MathPhysBioBooks/mathHistory/Boyer-AHistoryOfMathematics.pdf)
11. [Zeros of nonlinear functions, Universidad de Oviedo numerical methods lab](https://www.unioviedo.es/compnum/labs/lab04_ne/lab04_ne.html)
12. [ME 537 Numerical Methods in Engineering, Gaziantep University](https://akbis.gaziantep.edu.tr/yonetim/upload/files/38768-6012.pdf)
13. [Secant Method: Derivative-Free Root Finding, TheoremPath](https://www.theorempath.com/topics/secant-method)
14. [Numerical Analysis / Scientific Computing notes, University of Illinois](https://andreask.cs.illinois.edu/cs450-s26/notes.pdf)
15. [Numerical Methods, Bern University of Applied Sciences](https://web.sha1.bfh.science/NumMethods/NumMethods.pdf)
16. [Secant search, Dictionary of Algorithms and Data Structures, NIST](https://xlinux.nist.gov/dads/HTML/secantSearch.html)
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