A navigation instrument comprising: - a first sensor configured to measure three components of a first vector, called first measured vector, of a first terrestrial force field ( g ′ mes ⇀; B ′ mes ⇀) in a reference frame ( 1 x ′ ⇀, 1 y ′ ⇀, 1 z ′ ⇀) linked to said navigation instrument, called on-board reference frame, and generate first output signals which are representative of said three components ( g x, mes ′, g y, mes ′, g z, mes ′; B x, mes ′, B y, mes ′, B z, mes ′); - an orientation angle (φ, θ, ψ) calculation unit of the navigation instrument, this calculation unit being connected to said first sensor to receive said first output signals and being configured to be able to calculate a first orientation angle (φ) and a second orientation angle (θ) of said on-board reference frame, and thus of the navigation instrument, with respect to a terrestrial reference frame ( 1 x ⇀, 1 y ⇀, 1 z ⇀), these first and second orientation angles being defined by two successive rotations that the first measured vector must undergo, around a first reference axis of rotation ( 1 x ′ ⇀) of the on-board reference frame for the first orientation angle and a second reference axis of rotation ( 1 y ′ ⇀) of the on-board reference frame for the second orientation angle, to align this first measured vector on a calculation vector having, in the on-board reference frame, three calculation components which are identical to three reference components, in the terrestrial reference frame, of a first reference vector ( g ref ⇀; B ref ⇀) which is predefined or predetermined in the terrestrial reference frame; characterised in that the orientation angle calculation unit is configured to be able to determine the first and second orientation angles in a given order for said two successive rotations and also in reverse order; and in that the calculation unit is arranged to be able to choose between said given order and said reverse order, to calculate the first and second orientation angles, based on a comparison between an indicator of a risk of error, quantifying a risk of instability of said calculation unit during the determination of the first and second orientation angles, and a predetermined threshold, said indicator of the risk of error being determined by the navigation instrument according to at least two of said first output signals. - a first sensor configured to measure three components of a first vector, called first measured vector, of a first terrestrial force field ( g ′ mes ⇀; B ′ mes ⇀) in a reference frame ( 1 x ′ ⇀, 1 y ′ ⇀, 1 z ′ ⇀) linked to said navigation instrument, called on-board reference frame, and generate first output signals which are representative of said three components ( g x, mes ′, g y, mes ′, g z, mes ′; B x, mes ′, B y, mes ′, B z, mes ′); - an orientation angle (φ, θ, ψ) calculation unit of the navigation instrument, this calculation unit being connected to said first sensor to receive said first output signals and being configured to be able to calculate a first orientation angle (φ) and a second orientation angle (θ) of said on-board reference frame, and thus of the navigation instrument, with respect to a terrestrial reference frame ( 1 x ⇀, 1 y ⇀, 1 z ⇀), these first and second orientation angles being defined by two successive rotations that the first measured vector must undergo, around a first reference axis of rotation ( 1 x ′ ⇀) of the on-board reference frame for the first orientation angle and a second reference axis of rotation ( 1 y ′ ⇀) of the on-board reference frame for the second orientation angle, to align this first measured vector on a calculation vector having, in the on-board reference frame, three calculation components which are identical to three reference components, in the terrestrial reference frame, of a first reference vector ( g ref ⇀; B ref ⇀) which is predefined or predetermined in the terrestrial reference frame; characterised in that the orientation angle calculation unit is configured to be able to determine the first and second orientation angles in a given order for said two successive rotations and also in reverse order; and in that the calculation unit is arranged to be able to choose between said given order and said reverse order, to calculate the first and second orientation angles, based on a comparison between an indicator of a risk of error, quantifying a risk of instability of said calculation unit during the determination of the first and second orientation angles, and a predetermined threshold, said indicator of the risk of error being determined by the navigation instrument according to at least two of said first output signals.